Wave propagation and localization in a long-range correlated random potential

Abstract

We examine the effect of long-range spatially correlated disorder on the Anderson localization transition in $d=2+\ensuremath{\epsilon}$ dimensions. This is described as a phase transition in an appropriate non-linear $\ensuremath{\sigma}$ model. We consider a model of scalar waves in a medium with an inhomogeneous index of refraction characterized by scattering strength ${\ensuremath{\gamma}}^{2}$ and spatial correlations of range $a$ decaying (i) exponentially ${\ensuremath{\gamma}}_{1}^{2}{a}^{\ensuremath{-}d}{e}^{\ensuremath{-}\frac{x}{a}}$ and (ii) by power laws ${\ensuremath{\gamma}}_{2}^{2}{({a}^{2}+{x}^{2})}^{\ensuremath{-}m}(m>0)$. A replica-field-theory representation is utilized in the calculation of the one- and two-particle Green's functions. In addition to the usual diffusive Goldstone mode of the field theory arising from energy conservation, the non-linear $\ensuremath{\sigma}$ model is shown to possess a discrete spectrum of low-lying nondiffusive modes associated with approximate wave-vector ($\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$) conservation in the geometric optics limit $\mathrm{ka}\ensuremath{\gg}1$. For waves it is shown that all states are localized for $d\ensuremath{\le}2$ with diverging localization lengths in the low-frequency limit and that the mobility edge in $d=2+\ensuremath{\epsilon}$ separating high-frequency, localized states from low-frequency, extended states is characterized by the same critical exponents as for spatially uncorrelated disorder provided $m>\ensuremath{\epsilon}$. The problem of electron localization in a long-range correlated random potential is also described within the same universality class.