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

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.

arXiv comment52 pages, 27 Figures

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Cited by in corpus (13)

• Experimental observation of temporal pumping in electro-mechanical waveguides
• Observation of topological edge modes in a quasi-periodic acoustic waveguide
• Experimental Demonstration of Dynamic Topological Pumping Across Incommensurate Bilayered Acoustic Metamaterials
• Topological edge states of quasiperiodic elastic metasurfaces
• Topological Edge Modes by Smart Patterning
• Topological Gaps by Twisting
• Topological Gaps in Quasi-Periodic Spin Chains: A Numerical and K-Theoretic Analysis
• Revealing the boundary Weyl physics of the four-dimensional Hall effect via phason engineering in metamaterials
• Topological Lattice Defects by Groupoid Methods and Kasparov's KK-Theory
• Bulk-Boundary Correspondence for Topological Insulators with Quantized Magneto-Electric Effect
• Classifying the Dynamics of Architected Materials by Groupoid Methods
• Gap labeling theorem for multilayer thin film heterostructures
• Mapping Chern numbers in quasi-periodic interacting spin chains