Excitation spectrum for vibrations on a percolating network: Effective-medium approximation

Abstract

The excitation spectrum for vibrations on a (bond-) percolating network are calculated with the use of an effective-medium approximation. For $2<d<\mathrm{}4\mathrm{}$, where $d$ is the Euclidean dimensionality of the embedding space, we find a nearly linear relationship between frequency and wave vector for $\omega <{\omega }_c$, where ${\omega }_c$ represents the critical frequency separating phonon and fracton regimes as calculated previously by Derrida, Orbach, and Yu. The imaginary part of $\omega$ is small for $\omega <{\omega }_c$, signifying the correctness of a phonon eigenstate description in that regime. As the wave vector increases beyond the value corresponding to ${\omega }_c$, a plane-wave extended-state representation fails, signaled by a rapidly growing imaginary part of the frequency. It is interesting that an effective-medium approximation can sense the transition between extended and localized states. We calculate the $\omega$ dependence of what we characterize as the localization length $l\left(\omega \right)$. We find $\omega \sim l^{-2\mathrm{}}$ for $\omega <{\omega }_c$ in agreement with the scaling form generated by Alexander and Orbach. The length $l\left(\omega \right)$ diverges for $\omega <{\omega }_c$, as it should for wavelike excitations. Finally, we calculate the excitation spectrum for $1<d<\mathrm{}2\mathrm{}$, where Derrida et al. have shown that no sharp crossover occurs between phonon and fracton regimes. We expect both regimes to be localized. We find a smooth degradation of phonon character as $\omega$ increases, and a gradual transition to states with fracton character.