"Frames" consisting of nodes connected pairwise by rigid rods or central-force springs can model systems as diverse as architectural structures, crystalline and amorphous solids, granular matter, and protein structures. The rigidity of these networks depends on their average coordination number z: If z is small enough, the system has internal zero-frequency modes, and is "floppy"; if z is large enough, it is rigid. The critical point separating these two regimes is closely related to the so-called "isostatic" point, which for central forces in d-dimensions occurs at coordination number zc = 2d. At and near this rigidity threshold, elastic frames exhibit unique and interesting properties, including extreme sensitivity to boundary conditions, power-law scaling of elastic moduli, and diverging length and time scales. This talk will explore the properties of model periodic lattices, such as the square and kagome lattices with central-force springs, on the verge of mechanical instability. I will discuss the origin and nature of zero modes of these structures for both periodic and free boundaries, and derive general conditions (a) under which the zero modes in those two cases are essentially identical, and (b) under which phonon modes are gapped with no zero modes in the periodic case but include zero-frequency surface (Rayleigh) waves for free boundaries. In the former situation, lattices are generally in a type of critical state whose distortions give rise to surface modes whose penetration into the bulk diverges at criticality. The gapped states have a topological characterization, similar to that of topological insulators, which defines the nature of zero-modes at the boundary between systems with different topology.
Rigidity, Zero Modes, States of Self Stress, Topological States, and Surface Phonons in Periodic Networks at or near near their Instability Limit