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 kappa{m}(omega), equivalent to the complex shear modulus G(omega), obeys the scaling relation, kappa{m}(omega)=kappa{m}h(omega/omega{*}), at small P, where kappa{m}=kappa{m}{'}(0) approximately P{2} and omega{*} approximately P, implying nonaffine elastic response at small P and the breakdown of plane-wave states beyond the Ioffe-Regel limit at omega approximately omega{*}. We identify a divergent length l{*} approximately P{-1}, and we relate these results to jamming.