Fig. 1. (a) Schematic of two-dimensional magnetic photonic crystal fabricated by ferrite rods (denoted by yellow circles) embedded in air background. The black hexagon denotes the basis of the array. (b) Band structure at H = 0 and R = a/3. A four-fold degeneracy point shows at the Γ point. (c) Band structure at H ≠ 0 and R = a/3. A full bandgap presents in the band structure because of time reverse symmetry broken, and two doubly degeneracy points present at Γ point. (d) Band structure at H ≠ 0 and R ≠ a/3.

Fig. 2. Magnetic fields and the phase of electric fields for the eigenstates p₊, p_－, d₊, d_－ at the Г point. Reversing the direction of bias magnetic field causes the exchange in the profiles of electric phase and magnetic field.

Fig. 3. (a) Evolution of p and d eigenstates at point Г as a function of the radio of a/R with the bias magnetic field in the +z direction. As a/R increases, the band order exchange takes place where phase transition happens. Three types of phases are shaded in different colors. (b)−(d) Typical band structures of quantum spin-Hall (QSH) phase (R = a/2.684), quantum Hall (QH) phases (R = a/3), and conventional insulator (CI) phase (R = a/3.5). (e)−(h) The same as (a)−(d), but the bias magnetic field is in the –z direction. The insets of the panels (b)−(d) and (f)−(g) are the profile of eigenstates at the corresponding dots on the band structure of the panels, where the black arrow indicates the Poynting vectors.

Fig. 4. Influence of the direction of bias magnetic field on the edge states when two domains are in the QH phase but have different geometric structure (R = a/2.93 and R = a/3.09). (a) The projected band structure when two domains are applied by the same direction of bias magnetic field. (b) and (c) Electrical field distribution at 8 GHz excited by S₊ and S_–, respectively. No edge mode is excited. (d) The projected band structure when two domains are applied by the opposite direction of bias magnetic field. (e) and (f) Electric field distribution corresponding to (d) at 8 GHz excited by S₊ and S_–. The waves propagate unidirectional. The inset enlarges the field details which shows the orbital angular momentum of the wave. The inserts of (a) and (d) show the schematic of the relationship of the topological edge state and eigenstates in the two domains.

Fig. 5. (a) Schematics of the domain boundary created by two domains with distinct topological index. Supposing the domain II keeps in QSH phase. (b)−(d) The procedure of the topological edge state creation when domain I takes place the phase transiting from QSH phase to QH phase.

Fig. 6. Influence of the direction of bias magnetic field on the edge states when two domains are the QSH phase and the CI phases, respectively. (a), (b) The projected band structure for the two domains are respectively applied by the same and the opposite direction of bias magnetic field. The edge states localized at the boundary lead to a pseudo-spin dependent one-way propagation. The application of using opposite direction of bias magnetic field in the two domain reduces the gap of edge states. (c) Unidirectional wave propagation localized at domain interface excited by S₊. (d) Unidirectional wave propagation excited by S_–. (e) TE wave propagates along with opposite side when it is excited by S.

Fig. 7. (a) Schematic of the computational transmission efficiency. (b) The transmission efficiency corresponding to the Fig. 6. The transmission efficiency of the edge state under the condition of applying bias magnetic field with the same (red curve) or opposite direction (blue curve) in the two domains. The transmission efficiency of the full photonic crystals of the left domain is shown in green curve and the right domain in black curve. The shadow region indicates the common band gaps.