Engineering fragile topology in photonic crystals: Topological quantum chemistry of light

In recent years, there have been rapid advances in the parallel fields of electronic and photonic topological crystals. Topological photonic crystals in particular show promise for coherent transport of light and quantum information at macroscopic scales. In this work, we apply for the first time the recently developed theory of "Topological quantum chemistry" to the study of band structures in photonic crystals. This method allows us to design and diagnose topological photonic band structures using only group theory and linear algebra. As an example, we focus on a family of crystals formed by elliptical rods in a triangular lattice. We show that the symmetry of Bloch states in the Brillouin zone can determine the position of the localized photonic wave packets describing groups of bands. By modifying the crystal structure and inverting bands, we show how the centers of these wave packets can be moved between different positions in the unit cell. Finally, we show that for shapes of dielectric rods, there exist isolated topological bands which do not admit a well-localized description, representing the first physical instance of "fragile" topology in a truly noninteracting system. Our work demonstrates how photonic crystals are the natural platform for the future experimental investigation of fragile topological bands.