# Soft modes and elasticity of nearly isostatic lattices: randomness and dissipation - Condensed Matter > Soft Condensed Matter

Abstract: The square lattice with central-force springs on nearest-neighbor bonds isisostatic. It has a zero mode for each row and column, and it does not supportshear. Using the Coherent Potential Approximation CPA, we study how therandom addition, with probability $\mathcal{P}=z-4-4$ $z$ = average numberof nearest neighbors, of springs on next-nearest-neighbor $NNN$ bondsrestores rigidity and affects phonon structure. We find that the CPA effective$NNN$ spring constant $\tilde{\kappa} m\omega$, equivalent to the complexshear modulus $G\omega$, obeys the scaling relation,$\tilde{\kappa} m\omega = \kappa m h\omega-\omega^*$, at small$\mathcal{P}$, where $\kappa m = \tilde{\kappa}- m0\sim \mathcal{P}^2$ and$\omega^* \sim \mathcal{P}$, implying that elastic response is nonaffine atsmall $\mathcal{P}$ and that plane-wave states are ill-defined beyond theIoffe-Regel limit at $\omega\approx \omega^*$. We identify a divergent length$l^* \sim \mathcal{P}^{-1}$, and we relate these results to jamming.

Author: Xiaoming Mao, Ning Xu, T. C. Lubensky

Source: https://arxiv.org/