Fig. 1. Structure of the lattice. (a) Triangular compound lattice; (b) first Brillouin zone of the lattice and three high symmetry points

Fig. 2. Energy band and orbit of the initial structure. (a) r = 0.06a, D = 0.12a; (b) r = 0.06a, D = 0.44a

Fig. 3. Change curves of the energy band frequency with zooming distance D at different r. (a) r = 0.06a; (b) r = 0.057a; (c) r = 0.02a

Fig. 4. Curves of the frequency of the 3rd and 4th energy levels of the three high symmetry points with the dielectric cylinder radius. (a) Trivial state; (b) non-trivial state

Fig. 5. Width of the common band gap versus the radiuses of two structures

Fig. 6. Energy band after optimized structure parameters. (a) Trivial state; (b) non-trivial state

Fig. 7. Boundary state after optimization. (a) Supercell energy band diagram; (b) mode field at points A, B, C, and D in the boundary state mode and energy flux density vector at the boundary

Fig. 8. Transmission of the spin-locked boundary states. (a) Clockwise spin; (b) anticlockwise spin

Fig. 9. Bending transmission of the spin-locked boundary states. （a） Clockwise spin； （b） anticlockwise spin