Classifying the Dynamics of Architected Materials by Groupoid Methods

TL;DR

This paper introduces a mathematical framework using groupoid C*-algebras to classify the dynamical behaviors of architected materials with resonators, linking their structure to algebraic invariants.

Contribution

It provides a novel method to compute the minimal C*-algebra for these materials, connecting their architecture to algebraic classification.

Findings

C*-algebras classify materials by dynamical properties

Groupoid methods relate architecture to algebraic invariants

Materials with identical C*-algebras share dynamical behavior

Abstract

We consider synthetic materials consisting of self-coupled identical resonators carrying classical internal degrees of freedom. The architecture of such material is specified by the positions and orientations of the resonators. Our goal is to calculate the smallest C*-algebra that covers the dynamical matrices associated to a fixed architecture and adjustable internal structures. We give the answer in terms of a groupoid C*-algebra that can be canonically associated to a uniformly discrete subset of the group of isometries of the Euclidean space. Our result implies that the isomorphism classes of these C*-algebras split these architected materials into classes containing materials that are identical from the dynamical point of view.