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 $d=2+\epsilon$ dimensions. This is described as a phase transition in an appropriate non-linear $\sigma$ model. We consider a model of scalar waves in a medium with an inhomogeneous index of refraction characterized by scattering strength ${\gamma }^2$ and spatial correlations of range $a$ decaying (i) exponentially ${\gamma }_1^2{a}^{-d}{e}^{-\frac{x}{a}}$ and (ii) by power laws ${\gamma }_2^2{(a^2+x^2)}^{-m}\mathrm{}(m>\mathrm{}0\mathrm{})\mathrm{}$. 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 $\sigma$ model is shown to possess a discrete spectrum of low-lying nondiffusive modes associated with approximate wave-vector ($\overrightarrow{\mathrm{k}}$) conservation in the geometric optics limit $\mathrm{ka}\gg 1$. For waves it is shown that all states are localized for $d\le 2$ with diverging localization lengths in the low-frequency limit and that the mobility edge in $d=2+\epsilon$ separating high-frequency, localized states from low-frequency, extended states is characterized by the same critical exponents as for spatially uncorrelated disorder provided $m>\mathrm{}\epsilon$. The problem of electron localization in a long-range correlated random potential is also described within the same universality class.