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 $\text{Prob}=\mathrm{\Delta }z\mathrm{/}4$, where $\mathrm{\Delta }z=z-4$ and $z$ is the average coordination number. The effective medium static $NNN$ spring constant ${\kappa }_m$ scales as ${\text{Prob}}^2$ for $\text{Prob}\ll \kappa$ and as $\text{Prob}$ for $\text{Prob}\gg \kappa$, yielding a frequency scale ${\omega }^{\ast }\sim \mathrm{\Delta }z$ and a length scale $l^\sim (\Delta z)^{-1}$\mathrm{.}Toaverygoodapproximationatatsmallnonzerofrequency,$\kappa_m(\Prob,\omega)/\kappa_m(\Prob,0)$isascalingfunctionof$\omega/\omega^$. The Ioffe-Regel limit beyond which plane-wave states becomes ill-define is reached at a frequency of order ${\omega }^{\ast }$.