The K-Theoretic Bulk-Boundary Principle for Dynamically Patterned Resonators

Starting from a dynamical system $(\mathrm{\Omega },G)$, with $G$ a generic topological group, we devise algorithms that generate families of patterns in the Euclidean space, which densely embed $G$ and on which $G$ acts continuously by rigid shifts. We refer to such patterns as being dynamically generated. For $G={\mathbb{Z}}^d$, we adopt Bellissard’s $C^{\ast }$-algebraic formalism to analyze the dynamics of coupled resonators arranged in dynamically generated point patterns. We then use the standard connecting maps of $K$-theory to derive precise conditions that assure the existence of topological boundary modes when a sample is halved. We supply four examples for which the calculations can be carried explicitly. The predictions are supported by many numerical experiments.