Phonons and elasticity in critically coordinated lattices
arXiv:1503.01324
Abstract
Much of our understanding of vibrational excitations and elasticity is based upon analysis of frames consisting of sites connected by bonds occupied by central-force springs, the stability of which depends on the average number of neighbors per site . When , where is the spatial dimension, frames are unstable with respect to internal deformations. This pedagogical review focuses on properties of frames with at or near , which model systems like randomly packed spheres near jamming and network glasses. Using an index theorem, relating the number of sites, , and number of bonds, , to the number, , of modes of zero energy and the number, , of states of self stress, in which springs can be under positive or negative tension while forces on sites remain zero, it explores the properties of periodic square, kagome, and related lattices for which and the relation between states of self stress and zero modes in periodic lattices to the surface zero modes of finite free lattices (with free boundary conditions). It shows how modifications to the periodic kagome lattice can eliminate all but trivial translational zero modes and create topologically distinct classes, analogous to those of topological insulators, with protected zero modes at free boundaries and at interfaces between different topological classes.
References in corpus (44)
- Topological Insulators
- Topological insulators and superconductors
- Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
- Topological Order and the Quantum Spin Hall Effect
- Topological Insulators in Three Dimensions
- The World as a Hologram
- Topological invariants of time-reversal-invariant band structures
- Jamming at Zero Temperature and Zero Applied Stress: the Epitome of Disorder
- Topological Boundary Modes in Isostatic Lattices
- Modeling semiflexible polymer networks
- Weyl points and line nodes in gapless gyroid photonic crystals
- Random Packings of Frictionless Particles
- Effects of compression on the vibrational modes of marginally jammed solids
- Vibrations and diverging length scales near the unjamming transition
- Elasticity of Stiff Polymer Networks
- Topological modes bound to dislocations in mechanical metamaterials
- Geometric origin of excess low-frequency vibrational modes in amorphous solids
- Deformation of crosslinked semiflexible polymer networks
- Criticality and isostaticity in fiber networks
- On the Rigidity of Amorphous Solids
- Adler-Bell-Jackiw anomaly in Weyl semi-metals: Application to Pyrochlore Iridates
- Isostaticity, auxetic response, surface modes, and conformal invariance in twisted kagome lattices
- Finite-Size Scaling at the Jamming Transition
- A unified framework for non-Brownian suspension flows and soft amorphous solids
- Self-Organization and the Physics of Glassy Networks
- Floppy modes and non-affine deformations in random fiber networks
- Tunable Line Node Semimetals
- Excitonic Phases from Weyl Semi-Metals
- Nonaffine rubber elasticity for stiff polymer networks
- Soft Sphere Packings at Finite Pressure but Unstable to Shear
- Jamming in finite systems: stability, anisotropy, fluctuations and scaling
- Energy transport in jammed sphere packings
- Elasticity and Response in Nearly Isostatic Periodic Lattices
- Algorithms for 3D rigidity analysis and a first order percolation transition
- Soft modes and elasticity of nearly isostatic lattices: randomness and dissipation
- Phonon gap and localization lengths in floppy materials
- Elasticity of Filamentous Kagome Lattice
- Why glass elasticity affects the thermodynamics and fragility of super-cooled liquids
- Effect of contacts on spin lifetime measurements in graphene
- Ordered Loop Current States in Bilayer Graphene
- Long-range interaction induced phases in Weyl semimetals
- Stability of jammed packings II: the transverse length scale
- Magneto-Chiral Kerr effect with application to the cuprates
- Field-effect mobility enhanced by tuning the Fermi level into the band gap of Bi2Se3