Phonons and elasticity in critically coordinated lattices

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 $z$. When $z<z_c \approx 2d$, where $d$ is the spatial dimension, frames are unstable with respect to internal deformations. This pedagogical review focuses on properties of frames with $z$ at or near $z_c$, which model systems like randomly packed spheres near jamming and network glasses. Using an index theorem, $N_0 - N_S = dN - N_B$ relating the number of sites, $N$, and number of bonds, $N_B$, to the number, $N_0$, of modes of zero energy and the number, $N_S$, 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 $z=z_c$ 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.