Soft Modes and Elasticity of Nearly Isostatic Lattices: Randomness and Dissipation

The square lattice with nearest neighbor central-force springs is isostatic and does not support shear. Using the coherent potential approximation (CPA), we study how the random addition, with probability P=(z-4)/4 (z=average number of contacts), of next-nearest-neighbor (NNN) springs restores rigidity and affects phonon structure. The CPA effective NNN spring constant κₘ(ω), equivalent to the complex shear modulus G(ω), obeys the scaling relation, κₘ(ω)=κₘh(ω/ω*), at small P, where κₘ=κₘ^'(0)~P² and ω*~P, implying nonaffine elastic response at small P and the breakdown of plane-wave states beyond the Ioffe-Regel limit at ω≈ω*. We identify a divergent length l*~P^-1, and we relate these results to jamming.