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

TL;DR

This paper develops algorithms to generate dynamically embedded patterns in Euclidean space from a topological group and uses $K$-theory to analyze topological boundary modes in coupled resonator systems, supported by explicit examples and numerical experiments.

Contribution

It introduces a $K$-theoretic bulk-boundary principle for dynamically generated resonator patterns, extending topological analysis to new classes of aperiodic structures.

Findings

Conditions for topological boundary modes are derived.

Explicit calculations are provided for four example patterns.

Numerical experiments support theoretical predictions.

Abstract

Starting from a dynamical system $(\Omega,G)$, 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.