Coherent potential approximation of random nearly isostatic kagome lattice

The kagome lattice has coordination number $4$, and it is mechanically isostatic when nearest neighbor ($NN$) sites are connected by central force springs. A lattice of $N$ sites has $O(\sqrt{N})$ zero-frequency floppy modes that convert to finite-frequency anomalous modes when next-nearest-neighbor ($NNN$) springs are added. We use the coherent potential approximation (CPA) to study the mode structure and mechanical properties of the kagome lattice in which $NNN$ springs with spring constant $\kappa$ are added with probability $\Prob= \Delta z/4$, where $\Delta z= z-4$ and $z$ is the average coordination number. The effective medium static $NNN$ spring constant $\kappa_m$ scales as $\Prob^2$ for $\Prob \ll \kappa$ and as $\Prob$ for $\Prob \gg \kappa$, yielding a frequency scale $\omega^* \sim \Delta z$ and a length scale $l^*\sim (\Delta z)^{-1}$. To a very good approximation at at small nonzero frequency, $\kappa_m(\Prob,\omega)/\kappa_m(\Prob,0)$ is a scaling function of $\omega/\omega^*$. The Ioffe-Regel limit beyond which plane-wave states becomes ill-define is reached at a frequency of order $\omega^*$.