Fig. 1. (a) Left: Schematic of unrotated sampled PhC with lattice constant a0 where the three positions in the C3 point group are labeled by o, p, q, respectively. Right: Rotated unit cell with θ as the rotation angle. (b)–(d) Dispersion bands of the valley PhC with θ=−30°,0°, and 30° [insets of (b) and (d) show the phase distributions]. Valley points of lower and higher frequencies are labeled by K1 (K1′) and K2 (K2′), respectively. (b) When θ=−30°, at the p1 and q1 points the phase distributions reveal that K1 (K1′) and K2 (K2′) have opposite chiralities, whose handedness is indicated by the black arrow in the insets. (c) When θ=0°, the Dirac points appear at points K and K′ in the FBZ, and the inset shows the FBZ of the triangular lattice. (d) When θ=30°, K and K′ valley points near the bandgap are reversed in frequency order at q2 and p2 compared to panel (b). (e) The blue and the red bands represent the frequency gap variation of the K valley when the unit cell rotates between −60° and +60° in a period. The crayon and orange shadings indicate the complete bandgap width of bands when rotating for different angles. Note that no complete gaps remain between −4° and +4°.

Fig. 2. Electric-field distribution |Ez|(x,y) of the K-valley state (low-frequency K1, high-frequency K2) at positions p, q (p, q indicate positions with C3v symmetry). Parameter: rotational angle θ=30°. The upper and lower panels, respectively, represent the valley phase and electric-field amplitude distribution of the large-period lattice, and the arrows in the lower panel indicate the corresponding time-averaged Poynting vector.

Fig. 3. (a) Bulk polarization changes when the unit cells rotate synchronously. Red circles for P1, blue dots for P2, dotted line for the theoretical calculation, and the inset for the schematic of the FBZ. (b) Left: When θ=−30°, the polarization value along the wave-vector b1,2 direction P1,2=−1/3 [bulk polarization P=(−1/3,−1/3)] where the Wannier center in the unit cell aligns at the maximal Wyckoff positions p (blue dots in the inset). Right: When θ=30°, bulk polarization P=(1/3,1/3) where the Wannier center is located at the maximal Wyckoff position q (red dots in the inset).

Fig. 4. Up-corner states and down-corner states in a triangular nanodisk with opposite polarization. (a) Eigenfrequency evolution spectrum when θ∈(−30°,−10°). Red solid line for U-I corner states, and blue dot-dashed line for U-II corner states. (b) Left panel: Schematic for zigzag edges with the DPC surrounding the UPC. Right panel: The UPC eigenspectra of the bulk-edge-corner states where the blue dots on the UPC indicate the positions of the Wannier centers selected by U-I and U-II corner states. (c) Electric-field distribution |Ez|(x,y) of the U-I corner states at frequency f=5.9656  THz and of the U-II corner state at frequency f=6.0309  THz. (d) Eigenfrequency evolution spectra when θ∈(10°,30°). Red solid line for D-I corner states, and blue dot-dashed line for D-II corner states. (e) Left panel: Schematic for zigzag edges with the UPC surrounding the DPC. Right panel: The DPC eigenfrequency distribution of the bulk-edge-corner states. The red dot of the DPC superunit model represents the Wannier center selected by the down-corner states. (f) Electric-field distribution of D-I and D-II corner states at frequencies f=5.8301 and 6.4264 THz, respectively.

Fig. 5. Simulated electric-field |Ez|(x,y) for a configuration consisting of the UPC [cf. inset of Fig. 4(b)] and the DPC [cf. inset of Fig. 4(e)]. The blue pentagram at the bottom of the configuration indicates a chiral OAM source. (a) Field distributions of the U-I and U-II corners and U edge where the UPC is surrounded by the DPC are excited by left-handed circularly polarized (LCP) chiral sources with frequencies f=5.9650,6.0310,6.1700  THz. (b) Field distributions of D-I, D-II corners, and the D edge where the DPC is surrounded by the UPC are also excited by LCP sources with frequencies f=5.8301,6.4260,6.1400  THz.

Fig. 6. TM mode dispersion diagrams when the PhC unit cell rotates for different angles. (a)–(c) When θ=−30° and θ=30°, four complete bandgaps are produced, the green regions indicate the nontrival bandgap, and the brown regions indicate the trival bandgap. (b) When θ=0°, bands 2, 3, 7, and 8 are closed. Four Dirac points (phase-transition points) appear at K and K′, and the topological nontrivial bandgap disappears. (d) When θ=−30° and θ=30°, the phase distribution of bands 2 and 3 at points K and K′. (e) Berry curvature near K and K′ for the 7th band.

Fig. 7. (a) Schematic showing our choice of lattice vectors a1,a2 for C3 TCIs. (b) BZ and reciprocal lattice vectors for C3-symmetric crystals, b1=2π(1,−1/3),b2=2π(1,1/3). FBZ of crystals with C3 symmetries and their rotation invariant points; K and K′ are threefold HSPs. (c) Three maximal Wyckoff positions in the C3-symmetric: point o at the center of the unit cell and the points p, q at the corners of the unit cell.

Fig. 8. Dual-polarization models (UPC and DPC) select the corner state in the armchair boundary. (a) Left panel: Schematic structure for armchair edges with the DPC surrounding the UPC. Right panel: UPC eigenfrequency distribution of the bulk-edge-corner states. The red and blue dots indicate the positions of U-I and U-II corner states, and the brown dots indicate those of corner states. (b)–(d) Electric-field distribution of U-I and U-II corner states. (e) Left panel: Schematic for armchair edges with the UPC surrounding the DPC. Right panel: DPC eigenfrequency distribution of the bulk-edge-corner states. The red and blue dots indicate the positions of D-I and D-II corner states, and the brown dots indicate those of the corner states. (f)–(h) Electric-field distribution of D-I and D-II corner states. Note that the U-I corner states both appear at the same p positions in (b) and (c), and the D-I corner states appear at the same q positions in (g) and (h).

Fig. 9. Projection bands for the zigzag and armchair interfaces between the UPC and the DPC and its unit-cell layout. (a) The corner state of the zigzag interface appears below the edge state. (b) The corner state of the zigzag interface appears above and below the edge state. (c) and (d) The corner state of the armchair interface all appears below the edge state and the gray area is marked according to the frequency interval of corner states. Note that we omit two edge dispersion curves in (c) and (d) resulting from irrelevant interaction due to perfect electrical conductor (PEC) boundaries along the y direction.

Fig. 10. Defected waveguides of various shapes excited by an OAM source, which is an LCP OAM one. (a) In-line waveguide along the zigzag interface; (b)–(e) curved waveguides along the zigzag interface; (f) transmission spectrum for three zigzag edge states (a), (b), and (d) [cf. also the edge mode of Fig. 9(a)], which gives a wide bandwidth between 5.9 and 6.5 THz.