Coherent potential approximation of random nearly isostatic kagome lattice

Xiaoming Mao and T. C. Lubensky
Phys. Rev. E 83, 011111 – Published 18 January 2011

Abstract

The kagome lattice has coordination number 4, and it is mechanically isostatic when nearest-neighbor 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 (NNN) springs are added. We use the coherent potential approximation to study the mode structure and mechanical properties of the kagome lattice in which NNN springs with spring constant κ are added with probability P=Δz/4, where Δz=z4 and z is the average coordination number. The effective medium static NNN spring constant κm scales as P2 for Pκ and as P for Pκ, yielding a frequency scale ω*~Δz and a length scale l*~(Δz)1. To a very good approximation at small nonzero frequency, κm(P,ω)/κm(P,0) is a scaling function of ω/ω*. The Ioffe-Regel limit beyond which plane-wave states become ill-defined is reached at a frequency of order ω*.

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  • Received 11 August 2010

DOI:https://doi.org/10.1103/PhysRevE.83.011111

© 2011 American Physical Society

Authors & Affiliations

Xiaoming Mao and T. C. Lubensky

  • Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

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Issue

Vol. 83, Iss. 1 — January 2011

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