Isostaticity, auxetic response, surface modes, and conformal invariance in twisted kagome lattices

Abstract

Model lattices consisting of balls connected by central-force springs provide much of our understanding of mechanical response and phonon structure of real materials. Their stability depends critically on their coordination number $z$. $d$-dimensional lattices with $z=2d$ are at the threshold of mechanical stability and are isostatic. Lattices with $z<2d$ exhibit zero-frequency "floppy" modes that provide avenues for lattice collapse. The physics of systems as diverse as architectural structures, network glasses, randomly packed spheres, and biopolymer networks is strongly influenced by a nearby isostatic lattice. We explore elasticity and phonons of a special class of two-dimensional isostatic lattices constructed by distorting the kagome lattice. We show that the phonon structure of these lattices, characterized by vanishing bulk moduli and thus negative Poisson ratios and auxetic elasticity, depends sensitively on boundary conditions and on the nature of the kagome distortions. We construct lattices that under free boundary conditions exhibit surface floppy modes only or a combination of both surface and bulk floppy modes; and we show that bulk floppy modes present under free boundary conditions are also present under periodic boundary conditions but that surface modes are not. In the the long-wavelength limit, the elastic theory of all these lattices is a conformally invariant field theory with holographic properties, and the surface waves are Rayleigh waves. We discuss our results in relation to recent work on jammed systems. Our results highlight the importance of network architecture in determining floppy-mode structure.

arXiv comment12 pages, 7 figures

References in corpus (15)

• Topological Insulators
• Topological insulators and superconductors
• The World as a Hologram
• Jamming at Zero Temperature and Zero Applied Stress: the Epitome of Disorder
• Jammed Hard-Particle Packings: From Kepler to Bernal and Beyond
• Low-temperature properties of classical, geometrically frustrated antiferromagnets
• Vibrations and diverging length scales near the unjamming transition
• Elasticity of Stiff Polymer Networks
• Geometric origin of excess low-frequency vibrational modes in amorphous solids
• Criticality and isostaticity in fiber networks
• On the Rigidity of Amorphous Solids
• Floppy modes and non-affine deformations in random fiber networks
• Elasticity and Response in Nearly Isostatic Periodic Lattices
• Soft modes and elasticity of nearly isostatic lattices: randomness and dissipation
• Coherent potential approximation of random nearly isostatic kagome lattice

Cited by in corpus (104)

• Topological Phononic Crystals with One-Way Elastic Edge Waves
• Topological Boundary Modes in Isostatic Lattices
• Modeling semiflexible polymer networks
• Topological modes bound to dislocations in mechanical metamaterials
• Nonlinear conduction via solitons in a topological mechanical insulator
• Selective buckling via states of self-stress in topological metamaterials
• Transformable topological mechanical metamaterials
• Non-Hermitian band topology and skin modes in active elastic media
• Mechanical Weyl Modes in Topological Maxwell Lattices
• Topological phonons and Weyl lines in 3 dimensions
• Complete characterization of the macroscopic deformations of periodic unimode metamaterials of rigid bars and pivots
• Stiffest Elastic Networks
• Chiral Assemblies of Pinwheel Superlattices on Substrates
• Lattice Mechanics of Origami Tessellations
• Rigidity Loss in Disordered Systems: Three Scenarios
• Dualities and non-Abelian mechanics
• A discretized Chern-Simons gauge theory on arbitrary graphs
• Geared topological metamaterials with tunable mechanical stability
• Conformal Elasticity of Mechanism-Based Metamaterials
• Knotted Defects in Nematic Liquid Crystals
• Mechanics of elastic networks
• Silicene vs. ordered 2D silicide: the atomic and electronic structure of the Si-$(\sqrt{19}\times \sqrt{19})R23.4^{\circ }$/Pt(111) surface reconstruction
• Edge Modes and Asymmetric Wave Transport in Topological Lattices: Experimental Characterization at Finite Frequencies
• Topological edge floppy modes in disordered fiber networks
• Band theory and boundary modes of high-dimensional representations of infinite hyperbolic lattices
• Fracturing of topological Maxwell lattices
• Enumerating rigid sphere packings
• Elasticity of Filamentous Kagome Lattice
• Geometry and the onset of rigidity in a disordered network
• Mechanical instability at finite temperature
• Phonon analogue of topological nodal semimetals
• Chiral edge mode in the coupled dynamics of magnetic solitons in a honeycomb lattice
• Reconstruction of moiré lattices in twisted transition metal dichalcogenide bilayers
• Length scales and self-organization in dense suspension flows
• Microtwist elasticity: A continuum approach to zero modes and topological polarization in Kagome lattices
• Tunable Band Topology in Gyroscopic Lattices
• Universal Continuum Theory for Topological Edge Soft Modes
• Directional mechanical response in the bulk of topological metamaterials
• Self-assembly of three-dimensional open structures using patchy colloidal particles
• Topological mechanics in quasicrystals
• Buckling of thermalized elastic sheets
• Autonomous actuation of zero modes in mechanical networks far from equilibrium
• Localizing softness and stress along loops in three-dimensional topological metamaterials
• Geometric auxetics
• Topologically protected interface phonons in two-dimensional nanomaterials: hexagonal boron nitride and silicon carbide
• Rigidity percolation by next-nearest-neighbor braces on generic and regular isostatic lattices
• Theory of spin waves in a hexagonal antiferromagnet
• Simple views on symmetries and dualities in the theory of elasticity
• Theorem on the Compatibility of Spherical Kirigami Tessellations
• Topological boundary modes in jammed matter
• Entropic effects in the self-assembly of open lattices from patchy particles
• Mechanical Graphene
• Odd Cosserat elasticity in active materials
• Soft mechanical metamaterials with transformable topology protected by stress caching
• Existence of Corner Modes in Elastic Twisted Kagome Lattices
• Breakdown of Conventional Winding Number Calculation in One-Dimensional Lattices with Interactions Beyond Nearest Neighbors
• Symmetry detection of auxetic behaviour in 2D frameworks
• Deterministic and stochastic control of kirigami topology
• Shear shocks in fragile networks
• Hybrid topological magnon-phonon modes in honeycomb and kagome lattices
• Synthetically Non-Hermitian Nonlinear Wave-like Behavior in a Topological Mechanical Metamaterial
• Two-dimensional Mechanical Metamaterials with Unusual Poisson Ratio Behavior
• Symmetry of the phonon landscape across the duality boundary of twisted kagomes lattices
• Penrose Tilings as Jammed Solids
• Multi-functional Twisted-Kagome lattices: Tuning by Pruning Mechanical Metamaterials
• Algebraic non-Hermitian skin effect and generalized Fermi surface formula in arbitrary dimensions
• Self-Assembly of Isostatic Self-Dual Colloidal Crystals
• Scale without conformal invariance in membrane theory
• Topological Flexural Modes in Polarized Bilayer Lattices
• Topology and geometry of spin origami
• Some results on the Guest-Hutchinson modes and periodic mechanisms of the Kagome lattice metamaterial
• Finite temperature mechanical instability in disordered lattices
• Dynamics of self-dual kagome metamaterials and the emergence of fragile topology
• Finite-Frequency Topological Maxwell Modes in Mechanical Self-Dual Kagome Lattices
• Realizing negative Poisson's ratio in unstressed spring networks
• Disordered auxetic networks with no re-entrant polygons
• Discharging dynamics of topological batteries
• Duality,Hidden Symmetry and Dynamic Isomerism in 2D Hinge Structures
• Empty perovskites as Coulomb floppy networks: entropic elasticity and negative thermal expansion
• Rayleigh surface waves of extremal elastic materials
• Rayleigh Edge Waves in Two-Dimensional Crystals with Lorentz Forces -- from Skyrmion Crystals to Gyroscopic Media
• Patterns on a Roll: A Method for Continuous Feed Nanoprinting
• High Bulk Modulus Pentamodes: the Three-Dimensional Metal Water
• Capillary-driven binding of thin triangular prisms at fluid interfaces
• Mechanics of metric frustration in contorted filament bundles: From local symmetry to columnar elasticity
• Self-assembled multi-layer simple cubic photonic crystals of oppositely charged colloids in confinement
• Architecting mechanisms of damage in topological metamaterials
• Phonons and elasticity in critically coordinated lattices
• Omnidirectional domain wall modes protected by fragile topological states
• Ultrarigid periodic frameworks
• Curvature screening in draped mechanical metamaterial sheet
• Sparse phonon modes of a limit-periodic structure
• The first-order flexibility of a crystal framework
• Cell augmentation framework for topological lattices
• Non-local twist sequences in floppy kagome chains
• Localized interfacial Phonon Modes at the Electronic Axion Domain Wall
• Liftings and stresses for planar periodic frameworks
• Kagome modes, a new route to ultralow thermal conductivity?
• Transformable Super-Isostatic Crystals Self-Assembled from Segment Colloidal Rods
• Hierarchical Topological States without Dimension Reduction
• Classification of rotational zero modes in 2D micropolar solids
• Viscoelastic scaling regimes for marginally-rigid fractal spring networks
• Shape Entropy of a Reconfigurable Ising Surface
• Geometric approach to mechanical design principles in continuous elastic sheets