Fragile topology based helical edge states in two-dimensional moon-shaped photonic crystals

We study topological phases in a two-dimensional photonic crystal in a hexagonal lattice composed of an all-dielectric moon-shaped column (MSC) array, which preserves time-reversal symmetry, by calculating band structures, the Wilson loop, edge states, and topological-protected helical surface waves. Breaking the traditional way of changing the position or size of the cylinder, instead we simply rotate the MSCs to achieve topological phase transitions. Furthermore, the Wilson-loop method is used to directly calculate the topological properties of the bulk bands. By observing the eigenvalues of the Wilson loop, we demonstrate that bands in our system display fragile topology behavior. We also simulate the electromagnetic (EM) wave propagation in the designed photonic crystals with a point source at different operating frequencies, showing that the EM waves can propagate unidirectionally along the interface between topological trivial and nontrivial MSCs. Finally, the edge states at different frequencies can be easily tuned by adjusting the size of the MSCs. Our work provides a platform to study fragile topology and achieve frequency-tunable edge states, which may have potential applications for optical devices.