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Free, publicly-accessible full text available January 1, 2026
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Frieze groups are discrete subgroups of the full group of isometries of a flat strip. We investigate here the dynamics of specific architected materials generated by acting with a frieze group on a collection of self-coupling seed resonators. We demonstrate that, under unrestricted reconfigurations of the internal structures of the seed resonators, the dynamical matrices of the materials generate the full self-adjoint sector of the stabilized group C*-algebra of the frieze group. As a consequence, in applications where the positions, orientations and internal structures of the seed resonators are adiabatically modified, the spectral bands of the dynamical matrices carry a complete set of topological invariants that are fully accounted by the K-theory of the mentioned algebra. By resolving the generators of the K-theory, we produce the model dynamical matrices that carry the elementary topological charges, which we implement with systems of plate resonators to showcase several applications in spectral engineering. The paper is written in an expository style.
Free, publicly-accessible full text available June 1, 2025 -
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.more » « less
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Free, publicly-accessible full text available April 1, 2025
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The nontrivial topology of spin systems such as skyrmions in real space can promote complex electronic states. Here, we provide a general viewpoint at the emergence of topological spectral gaps in spin systems based on the methods of noncommutative-theory. By realizing that the structure of the observable algebra of spin textures is determined by the algebraic properties of the noncommutative torus, we arrive at a unified understanding of topological electronic states which we predict to arise in various noncollinear setups. The power of our approach lies in an ability to categorize emergent topological states algebraically without referring to smooth real- or reciprocal-space quantities. This opens a way towards an educated design of topological phases in aperiodic, disordered, or nonsmooth textures of spins and charges containing topological defects.
Published by the American Physical Society 2024