Nonabelian elastic metamaterials using holonomies acquired by crossing degeneracies

TL;DR

This paper introduces a novel design principle for continuous nonabelian elastic metamaterials that utilize crossing degeneracies to produce nonabelian holonomies, enabling advanced wave control in elastic media.

Contribution

It presents a new approach to realize nonabelian waveguides in continuous elastic media using crossing degeneracies, expanding beyond resonator-based systems.

Findings

Holonomy transfers excitation between rods.

Order of concatenated waveguides affects output, demonstrating nonabelian dynamics.

Robustness of nonabelian behavior across frequencies and perturbations.

Abstract

Embedding nonabelian features into elastic metamaterials promises remarkable opportunities for wave control in many practical applications such as surface acoustic wave devices, mode multiplexers, and on-material computation. Nevertheless, current realizations are limited to arrangements of coupled resonators with fine-tuned interactions, limiting their applicability to continuous media. This theoretical and numerical study introduces a design principle for continuous nonabelian elastic metamaterial waveguides. The basic configuration consists of a composite waveguide made of multiple cylindrical waveguides coupled by spatially varying elements. These elements are engineered to follow geometrically-controlled parameter variations that cross selected degeneracies and produce a targeted nonabelian holonomy. The strategy based on crossing degeneracies fundamentally differs from abelian geometric phases, where parameters avoid and encircle degeneracies, or nonabelian Wilczek-Zee phases, where parameters are fine-tuned to maintain degeneracies throughout their cycle. The resulting holonomy transfers an input longitudinal excitation in one rod to an output response in another rod. When two such waveguides are concatenated, their ordering dictates the output response, thereby revealing the emergence of nonabelian dynamics. The nonabelian behavior persists across a broad range of frequencies and under perturbations to the geometry of coupling elements or cylinder diameters. These results establish a robust, effective, and practical route to leverage nonabelian physics in elastic metamaterials.