Localization in a disordered elastic medium near two dimensions

Abstract

We describe the transition from extended to localized modes in a disordered elastic medium in $2+\epsilon$ dimensions as a phase transition in an appropriate nonlinear $\sigma$ model. The latter is derived by considering fluctuations about the mean-field approximation to the replica field Lagrangian for the system. Within this framework, we calculate the averaged one- and two-particle phonon Green's functions obtaining the phonon density of states and frequency-dependent, zero-temperature thermal diffusivity, respectively. Momentum-shell integration of the nonlinear $\sigma$ model reveals how this diffusivity renormalizes at longer length scales and hence the nature of normal modes at a given frequency. We demonstrate that all finite-frequency phonons in one and two dimensions are localized with low-frequency localization lengths diverging as $\frac{1}{{\omega }^2}$ and $e^{\frac{1}{{\omega }^2}}$, respectively, and that a mobility edge, ${\omega }_{*}$, separating low-frequency extended states from high-frequency localized states exists above $d=2\mathrm{}$. The phonon localization length at this mobility edge is shown to diverge as ${|\omega -{\omega }_{*}|}^{-\frac{1}{\epsilon }}$.