Topological Characterization of Photonic Crystals

We apply two of the most popular tools used in condensed matter physics to the topological characterization of dispersion bands in photonic crystals. We focus on a family of crystals formed by 2D cylinders placed as concentric hexagons in a triangular lattice. Applying the recently developed theory of "Topological Quantum Chemistry" [1] , we show that the symmetry of the Bloch’s wave functions can be used to determine the position of localized photonic Wannier functions — which can be moved in the unit cell by engineered band inversions. We can show this movement performing a discrete calculation of Wilson loops along one k -direction in the Brillouin zone [2] . Additionally, there is an isolated set of bands which does not admit a well-localized Wannier description, i.e topological bands. This represents a physical instance of "fragile" topology in photonic crystals [3] . Finally, we explore the local density of states (LDOS), and density of states (DOS) [4] . Through this analysis, we can analyze the distribution of the density of states for sets of isolated bands to conclude that the positions of their maximum concentration are related to the localization of maximally localized photonic Wannier functions. We expect that this approach will help the community to understand topological effects in photonic crystals in a more intuitive way.