Coherent potential approximation of random nearly isostatic kagome lattice

Xiaoming Mao; T. C. Lubensky

doi:10.1103/PhysRevE.83.011111

paper

Coherent potential approximation of random nearly isostatic kagome lattice

arXiv:1008.2037 · doi:10.1103/PhysRevE.83.011111

Abstract

The kagome lattice has coordination number $4$ , and it is mechanically isostatic when nearest neighbor ( $N N$ ) sites are connected by central force springs. A lattice of $N$ sites has $O (N)$ zero-frequency floppy modes that convert to finite-frequency anomalous modes when next-nearest-neighbor ( $N N N$ ) springs are added. We use the coherent potential approximation (CPA) to study the mode structure and mechanical properties of the kagome lattice in which $N N N$ springs with spring constant $κ$ are added with probability $\Prob= Δz/4$, where $Δ z = z - 4$ and $z$ is the average coordination number. The effective medium static $N N N$ spring constant $κ_{m}$ scales as $\Prob^2$ for $\Prob \ll κ$ and as $\Prob$ for $\Prob \gg κ$, yielding a frequency scale $ω^{*} \sim Δ z$ and a length scale $l^{*} \sim (Δ z)^{- 1}$ . 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 $ω^{*}$ .

15 pages, 8 figures

Coherent potential approximation of random nearly isostatic kagome lattice

Abstract

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