A new approach to the analytic solution of the Anderson localization problem for arbitrary dimensions

TL;DR

This paper introduces a new analytical method for determining the phase diagram of Anderson localization across arbitrary dimensions, using signal theory and the concept of generalized diffusion to distinguish between localized and delocalized states.

Contribution

It presents a novel analytical approach employing signal theory and generalized diffusion concepts to analyze Anderson localization in any spatial dimension.

Findings

The transition is characterized by the divergence of wavefunction correlators.

The Lyapunov exponent determines the localization length.

The filter function H(z) uniquely defines the transition point.

Abstract

Subsequent to the ideas presented in our previous papers [J.Phys.: Condens. Matter {\bf 14} (2002) 13777 and Eur. Phys. J. B {\bf 42} (2004) 529], we discuss here in detail a new analytical approach to calculating the phase-diagram for the Anderson localization in arbitrary spatial dimensions. The transition from delocalized to localized states is treated as a generalized diffusion which manifests itself in the divergence of averages of wavefunctions (correlators). This divergence is controlled by the Lyapunov exponent $\gamma$, which is the inverse of the localization length, $\xi=1/\gamma$. The appearance of the generalized diffusion arises due to the instability of a fundamental mode corresponding to correlators. The generalized diffusion can be described in terms of signal theory, which operates with the concepts of input and output signals and the filter function. Delocalized states correspond to bounded output signals, and localized states to unbounded output signals, respectively. Transition from bounded to unbounded signals is defined uniquely be the filter function $H(z)$. Simplifications in the mathematical derivations of the previous papers (averaging over initial conditions) are shown to be mathematically rigorous shortcuts.