Fig. 1. Schematic of primitive cell structures. (a) Hexagonal honeycomb lattice with C_6v symmetry; (b) lattice primitive cell; (c) the first Brillouin zone

Fig. 2. Structures of lattice a and lattice b. (a) Lattice a; (b) bands of lattice a with two Dirac cones at the valley point K; (c) lattice b; (d) low bands of Fig. 2(c) with two split valley points K⁺ and K^- and their phase distributions of E_z marked by arrows; (e) high bands of Fig. 2(c) with two split valley points K⁺ and K^- and their phase distributions of E_z marked by arrows

Fig. 3. Structures of lattice c and its bands. (a) Lattice c; (b) high bands of Fig. 3(a) with two split valley points K⁺ and K^-, and their phase distributions of E_zmarked by arrows; (c) low bands of Fig. 3(a) with two split valley points K⁺ and K^-, and their phase distributions of E_z marked by arrows

Fig. 4. Berry curvatures for lattice b and lattice c. (a) The first hexagon Brillouin zone for the calculations of Berry curvatures; (b) Berry curvature distribution for lattice b at low band gap; (c) Berry curvature distribution for lattice c at low band gap; (d) K point in the enlarged low band gap; (e) Berry curvature distribution for lattice b at high band gap; (f) Berry curvature distribution for lattice c at high band gap

Fig. 5. Model of type b-c-b super cell with its nontrivial band gaps and the energy flow vectors around the edge. (a) Model of type b-c-b super cell; (b) energy band of the super cell; (c) anticlockwise energy flow vectors at point P; (d) clockwise energy flow vectors at point P′

Fig. 6. Unidirectional verification. (a) High edge state with clockwise source and frequency at point P; (b) high edge state with clockwise source and frequency at point P′; (c) low edge state with clockwise source and frequency at point Q; (d) low edge state with clockwise source and frequency at point Q′

Fig. 7. Model design of nested photonic loops based on valley topological boundary waveguides

Fig. 8. Light flows from different positions or frequencies in the low edge state of nested photonic loops with normalized angular frequencies. (a) ω=0.86130(2πc/a); (b) ω=0.85820(2πc/a); (c) ω=0.86170(2πc/a); (d) ω=0.85120(2πc/a); (e) ω=0.86293(2πc/a); (f) ω=0.86293(2πc/a); (g) ω=0.86300(2πc/a); (h) ω=0.86300(2πc/a); (i) ω=0.86060(2πc/a); (j) ω=0.86200(2πc/a); (k) ω=0.86299(2πc/a)

Fig. 9. Light flows from different positions or frequencies in the high edge state of nested photonic loops with normalized angular frequencies. (a) ω=1.09310(2πc/a); (b) ω=1.09301(2πc/a); (c) ω=1.09307(2πc/a); (d) ω=1.07790(2πc/a); (e) ω=1.07200(2πc/a)