Topological Spectral Bands with Frieze Groups

Abstract

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 $C^{\ast }$-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.

References in corpus (27)

• Topological Crystalline Insulators
• Topological quantum chemistry
• The (High Quality) Topological Materials In The World
• Symmetry-based Indicators of Band Topology in the 230 Space Groups
• The Non-Commutative Geometry of the Quantum Hall Effect
• Inversion Symmetric Topological Insulators
• Topology of crystalline insulators and superconductors
• Magnetic Topological Quantum Chemistry
• Building Blocks of Topological Quantum Chemistry: Elementary Band Representations
• Twisted equivariant matter
• All Topological Bands of All Nonmagnetic Stoichiometric Materials
• Topological Materials Discovery from Crystal Symmetry
• Topology of nonsymmorphic crystalline insulators and superconductors
• Wallpaper Fermions and the Nonsymmorphic Dirac Insulator
• Non Commutative Geometry of Tilings and Gap Labelling
• Topological Crystalline Materials - General Formulation, Module Structure, and Wallpaper Groups -
• Graph Theory Data for Topological Quantum Chemistry
• Generalized homology and Atiyah-Hirzebruch spectral sequence in crystalline symmetry protected topological phenomena
• Topological Classification Table Implemented with Classical Passive Meta-Materials
• The structure of crossed products of irrational rotation algebras by finite subgroups of SL_2 (Z)
• Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups
• Twisted crystallograpic T-duality via the Baum--Connes isomorphism
• Crystallographic T-duality
• Crystallographic bulk-edge correspondence: glide reflections and twisted mod 2 indices
• Atiyah-Hirzebruch spectral sequence for topological insulators and superconductors: $E_2$ pages for 1651 magnetic space groups
• Fredholm modules over certain group C*-algebras
• Classifying the Dynamics of Architected Materials by Groupoid Methods