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 $κ$ are added with probability $\Prob= Δz/4$, where $Δz= z-4$ and $z$ is the average coordination number. The effective medium static $NNN$ spring constant $κ_m$ scales as $\Prob^2$ for $\Prob \ll κ$ and as $\Prob$ for $\Prob \gg κ$, yielding a frequency scale $ω^* \sim Δz$ and a length scale $l^*\sim (Δz)^{-1}$. To a very good approximation at at small nonzero frequency, $κ_m(\Prob,ω)/κ_m(\Prob,0)$ is a scaling function of $ω/ω^*$. The Ioffe-Regel limit beyond which plane-wave states becomes ill-define is reached at a frequency of order $ω^*$.

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