Coherent potential approximation of random nearly isostatic kagome lattice

TL;DR

This paper uses the coherent potential approximation to analyze how adding next-nearest-neighbor springs affects the vibrational modes and mechanical properties of a nearly isostatic kagome lattice, revealing scaling behaviors and the Ioffe-Regel limit.

Contribution

It introduces a CPA-based analysis of the kagome lattice with NNN springs, uncovering scaling laws for effective spring constants and vibrational frequency scales.

Findings

Effective medium spring constant scales as Prob^2 for small Prob and Prob for large Prob.

Frequency scale omega* is proportional to the excess coordination Delta z.

Ioffe-Regel limit occurs at frequencies around omega*.

Abstract

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 $\omega^*$.