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

The excitation spectrum for vibrations on a (bond-) percolating network are calculated with the use of an effective-medium approximation. For 2<d<4, where d is the Euclidean dimensionality of the embedding space, we find a nearly linear relationship between frequency and wave vector for <c, where c represents the critical frequency separating phonon and fracton regimes as calculated previously by Derrida, Orbach, and Yu. The imaginary part of is small for <c, signifying the correctness of a phonon eigenstate description in that regime. As the wave vector increases beyond the value corresponding to 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 dependence of what we characterize as the localization length l(). We find l-2 for <c in agreement with the scaling form generated by Alexander and Orbach. The length l() diverges for <c, as it should for wavelike excitations. Finally, we calculate the excitation spectrum for 1<d<2, 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 increases, and a gradual transition to states with fracton character.