Fig. 1. (a) Schematic of the imbalanced energy transport of the chiral edge states under the coexistence of the real magnetic field B0 and pseudomagnetic field Bs. (b) Haldane model and GPC configuration. The YIG cylinders are placed in a parallel metallic plate waveguide. Onsite magnetization modulation is realized by placing permanent magnets on the top and bottom of the cylinders. (c) Uniaxially strained GPC structure with a linear lattice deformation along the y axis producing the pseudomagnetic field along the z-direction. The background color indicates the distortion β quantifying the lattice deformation. Right panel: deformation of the honeycomb unit cells under stretching (red) and compression (blue). (d) Shift of the opened Dirac cones along the K-K′ direction for the stretched and compressed GPC. (e) Function relationship between lattice deviation mβ and Dirac point shift denoted by Δkx.

Fig. 2. (a) Schematic diagram of the experimental setup to observe the chiral edge state under the pseudomagnetic field. The gyromagnetic cylinders with a radius of 1.5 mm and height of 5 mm are placed within a waveguide consisting of parallel aluminum plates. Permanent magnets of radius 1.5 mm and height 0.5 mm placed exactly above and below the gyromagnetic cylinders provide a uniform external real magnetic field. (b) Top view of the experimental setup with deformed lattices of 7 layers corresponding to the distortion β of 0.4 mm. (c) Deformed band dispersion of the chiral edge states obtained from numerical simulations and experimental measurements. Upper and lower edge states are denoted by red and blue lines, respectively. (d) Numerically simulated edge state dispersion under different pseudomagnetic field intensities. (e) Left panel: |Ez| eigenfield distributions of the n=0, −1, and −2 Landau levels at the K′ valley with the frequencies of 9.08, 9.04, and 9 GHz and kx=0.34(2π/a). Right panel: eigenfields of the Landau levels at the K valley with the frequencies of 9.08, 9.04, and 9 GHz and kx=0.66(2π/a). Middle panel: normalized |Ez| as a function of the y coordinate at n=0, −1, and −2 Landau levels.

Fig. 3. (a) and (b) Simulated and experimentally measured electric field distribution of the imbalanced edge state transport. The electric field intensity at the upper edge is stronger compared to the lower edge at 9.25 GHz. Yellow stars mark excitation source positions, and blue stars indicate probe locations. (c) and (d) Simulated and measured transmission spectra.

Fig. 4. (a) Spatially asymmetric distribution of the antichiral edge states under the coexistence of the real magnetic field B0 and the pseudomagnetic field Bs. (b) Modified Haldane model and GPC configuration. (c) Deformed band structure of the antichiral edge states in numerical simulations and experimental measurements. (d) Band dispersion curves under different Bs intensities. (e) Left panel: |Ez| distributions of the n=0,−1, and −2 Landau levels at the K′ valley with the frequencies of 9.01, 8.95, and 8.85 GHz and kx=0.34(2π/a). Right panel: |Ez| distributions of the Landau levels for the K valley with the frequencies of 9.39, 9.33, and 9.26 GHz and kx=0.66(2π/a). Middle panel: normalized |Ez| as a function of the y coordinate at n=0, −1, and −2 Landau levels.

Fig. 5. (a) and (b) Simulated and experimentally measured electric field distribution of the antichiral edge states at 9.18 and 9.2 GHz, respectively. The yellow and blue stars indicate the position of the excited and probed sources, respectively. (c) and (d) Simulated and experimentally measured transmission spectra at P1, P2, and P3. (e) Electric field distribution of the bulk state excited by the excitation source indicated by the black arrow. (f) S21 and S12 transmission coefficients of the bulk states.

Fig. 6. (a) Schematic diagram of the five-channel unidirectional waveguide composed of four types of GPC structures under the interplay of the real magnetic field B0 and pseudomagnetic field Bs. The position of the excitation source is marked by a red star, and the transmission channels of electromagnetic waves are marked by red arrows. (b) Simulated electric field distribution of chiral and antichiral edge states at 9.34 GHz. (c) Chiral edge state dispersions of two GPCs with opposite Bs when the number of layers is 7 and the distortion β is 0.4 mm. Right panel: |Ez| eigen-electric field distribution at the frequency 9.24 GHz and the wave vector kx=0.414(2π/a) and 0.466(2π/a). Upper and lower edge states are denoted by red lines, and the middle channel is denoted by blue lines. (d) Antichiral edge state dispersions of two GPCs with opposite Bs. Right panel: |Ez| electric fields at the frequency 9.31 GHz and the wave vector kx=0.434(2π/a) and 0.532(2π/a).

Fig. 7. (a) Schematic of the GPCs with the Haldane model and chiral edge state dispersions within the photonic bandgap. (b) Antichiral edge state in the modified Haldane model.

Fig. 8. (a) and (c) Schematic diagram of the five-channel unidirectional waveguide composed of four types of GPC structures under the interplay of the real magnetic field B0 and pseudomagnetic field Bs. (b) and (d) Electric field distributions at 9.22 GHz and 9.23 GHz corresponding to the structures of (a) and (c), respectively.