Fig. 1. Geometric transitions in 2D PhCs with C3 symmetry. The primitive cells are indicated by hexagonal dotted lines with the lattice constant a and the side length l. A tunable parameter d with a range of 0–3l (the parameter d is modulo 3l) is employed to illustrate the geometric transitions between triangular, kagome, and breathing kagome configurations. By tuning the geometric parameter d, the C3 symmetry is preserved, while various configurations can be generated, including: (a) triangular I with d=0/d=3l, (b) kagome I with d=0.5l, (c) triangular II with d=l, (d) kagome II with d=1.5l, (e) triangular III with d=2l, and (f) kagome III with d=2.5l. Each primitive cell consists of three dielectric rods (possibly overlapping with each other) with identical radii r=0.1a and permittivity ϵ=15. (g) Possible position of the Wannier center for C3 symmetric unit cells.

Fig. 2. Photonic band structures of 2D PhCs with C3 symmetry for: (a) d=0,l,2l (i.e., triangular I, II, and III lattices), (b) d=0.5l,1.5l,2.5l (i.e., kagome I, II, and III lattices), and (c) d=0.25l,1.25l,2.25l (i.e., breathing kagome lattices). (d) The eigenfrequencies of the first and second photonic bands at the K point as functions of d. Band gaps of distinct topology are painted with different colors. The topological index χ is labeled for each region. The Wannier center for each region also is depicted. Insets illustrate the phase distributions of the eigenstates of the first photonic band at the K point for various d.

Fig. 3. (a) Schematic illustration of the large triangular supercells with two types of PhCs. The outer PhC has d=0.25l, while the inner PhC has variable d. Several cases with different d are shown in (a). (b) Eigenfrequencies of the photons as functions of the geometry parameter d. The gray regions represent the bulk states, the green regions represent the edge states, and the purple and blue curves represent the type-I and type-II corner states, respectively. (c) Electric field patterns of corner states and edge states with different d. Throughout this paper, the electric field patterns of the corner states are given by the superposition of |Ez| on the three degenerate corner states. In the calculation, the side length of the supercell is 10a, while the inside structure has a side length of 4a.

Fig. 4. (a) Schematic illustration of the large triangular supercells with two types of PhCs. The inner PhC has d=0.25l, while the outer PhC has variable d. Several cases with different d are shown in (a). (b) Eigenfrequencies of the photons as functions of the geometry parameter d. The gray regions represent the bulk states, the green regions represent the edge states, and the purple and blue curves represent the type-I and type-II corner states. (c) Electric field patterns of the corner states and edge states with different d. Throughout this paper, the electric field patterns of the corner states are given by the superposition of |Ez| on the three degenerate corner states. In the calculation, the side length of the supercell is 10a, while the inside structure has a side length of 4a.

Fig. 5. (a) Schematic illustration of the large triangular supercells with two types of PhCs. The outer PhC has the displacement d, while the inner PhC has the displacement 3l−d. Several cases with different d are shown in (a). (b) Eigenfrequencies of the photons as functions of the geometry parameter d. The gray regions represent the bulk states, the green regions represent the edge states, and the purple and blue curves represent the type-I and type-II corner states, respectively. (c) Electric field patterns of corner states and edge states with different d. Throughout this paper, the electric field patterns of the corner states are given by the superposition of |Ez| on the three degenerate corner states. In the calculation, the side length of the supercell is 10a, while the inside structure has a side length of 4a.

Fig. 6. Fractional “charges” in the triangular supercell with perfect electric conductor boundary conditions. Only the charges of the bulk unit cells are shown in the figure. The charges are calculated by including only the contributions from the bulk states below the topological gap, as indicated by the light blue areas. Four cases are considered: (a) d=0.25l, (b) d=0.75l, (c) d=2.75l, and (d) d=2.25l.

Fig. 7. (a) Adopted rhombic Brillouin zone in the calculation of the bulk polarization, which shares the same area with the original hexagonal Brillouin zone. The calculated Berry phase θ2,k1 as a function of k1 is presented, respectively, for the PhC with: (b) the phase χ1=[0,0], (c) the phase χ2=[−1,1], and (d) the phase χ3=[−1,0]. The bulk polarization is accordingly obtained by the integration of θ2,k1 over k1 and is shown by the red dotted lines (for illustration) with the exact values of 0, 1/3, and −1/3. The corresponding Wannier centers are marked by the black, blue, and green dots, which coincide with the Wyckoff positions. Note that the kagome PhCs with different configurations may share the same Wannier center positions.

Fig. 8. (a) Schematic illustration of the finite triangular-shaped supercell, where the intracell (intercell) coupling and next nearest neighbor hopping are denoted by t0(t1=10t0) and t2, respectively. The boundary is indicated by the red dotted line. (b) The eigenenergy of the finite triangular-shaped supercell versus the next nearest neighbor hopping strength. The green-gray regions represent the bulk states, the gray regions represent the edge states, and the red (purple) curves represent the type-I (type-II) corner states. (c) The eigenmodes of both edge states (“A” and “E”) and corner states (“B,” “C,” and “D”) with t2=0.5t0. In the calculation, the side length of the supercell is 16, while the inside structure has a side length of 14 (the lattice constant in the tight-binding model is set to unity).