Soft modes and elasticity of nearly isostatic lattices: randomness and dissipation

Abstract

The square lattice with central-force springs on nearest-neighbor bonds is isostatic. It has a zero mode for each row and column, and it does not support shear. Using the Coherent Potential Approximation (CPA), we study how the random addition, with probability $\mathcal{P}=(z-4)/4$ ($z$ = average number of nearest neighbors), of springs on next-nearest-neighbor ($NNN$) bonds restores rigidity and affects phonon structure. We find that the CPA effective $NNN$ spring constant ${\tilde{\kappa }}_m(\omega )$, equivalent to the complex shear modulus $G(\omega )$, obeys the scaling relation, ${\tilde{\kappa }}_m(\omega )={\kappa }_{m}h(\omega /{\omega }^{\ast })$, at small $\mathcal{P}$, where ${\kappa }_m={\tilde{\kappa }}_m^{\prime }(0)\sim {\mathcal{P}}^2$ and ${\omega }^{\ast }\sim \mathcal{P}$, implying that elastic response is nonaffine at small $\mathcal{P}$ and that plane-wave states are ill-defined beyond the Ioffe-Regel limit at $\omega \approx {\omega }^{\ast }$. We identify a divergent length $l^{\ast }\sim {\mathcal{P}}^{-1}$, and we relate these results to jamming.

4 pages, 4 figures

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