Local markers for crystalline topology

Over the last few years, crystalline topology has been used in photonic crystals to realize edge- and corner-localized states that enhance light-matter interactions for potential device applications. However, the band-theoretic approaches currently used to classify bulk topological crystalline phases cannot predict the existence, localization, or spectral isolation of any resulting boundary-localized modes. While interfaces between materials in different crystalline phases must have topological states at some energy, these states need not appear within the band gap, and thus may not be useful for applications. Here, we derive a class of local markers for identifying material topology due to crystalline symmetries, as well as a corresponding measure of topological protection. As our real-space-based approach is inherently local, it immediately reveals the existence and robustness of topological boundary-localized states, yielding a predictive framework for designing topological crystalline heterostructures. Beyond enabling the optimization of device geometries, we anticipate that our framework will also provide a route forward to deriving local markers for other classes of topology that are reliant upon spatial symmetries.