Ariel Amir, Jacob J. Krich, Vincenzo Vitelli, Yuval Oreg, Yoseph Imry
We study, theoretically and numerically, a minimal model for phonons in a disordered system that shows rich behavior in the localization properties of the phonons as a function of the density, frequency and the spatial dimension. We use a percolation analysis to argue for a Debye spectrum at low frequencies for dimensions higher than one, and for a localization/delocalization transition. We show that in contrast to the behavior in electronic systems, the transition exists for arbitrarily large disorder, albeit with an exponentially small critical frequency. The structure of the modes reflects a divergent percolation length that arises from the disorder in the springs without being explicitly present in the de?nition of our model. We calculate the speed-of-sound of the delocalized modes (phonons) and corroborate it with numerics. We calculate the critical frequency of the localization transition at a given density and test the prediction numerically using a recursive Green function method.
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http://arxiv.org/abs/1209.2169
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