Exact analytic solution for the generalized Lyapunov exponent of the 2-dimensional Anderson localization

TL;DR

This paper provides an exact analytical solution for the generalized Lyapunov exponent in 2D Anderson localization, revealing detailed phase behavior and the nature of the metal-insulator transition.

Contribution

It introduces a novel analytical method using signal theory to solve the Anderson localization problem in 2D, offering new insights into phase diagrams and transition types.

Findings

All states are localized in 1D for any disorder.

In 2D, extended and localized states coexist at certain energies and low disorder.

The metal-insulator transition is characterized as a first-order phase transition.

Abstract

The Anderson localization problem in one and two dimensions is solved analytically via the calculation of the generalized Lyapunov exponents. This is achieved by making use of signal theory. The phase diagram can be analyzed in this way. In the one dimensional case all states are localized for arbitrarily small disorder in agreement with existing theories. In the two dimensional case for larger energies and large disorder all states are localized but for certain energies and small disorder extended and localized states coexist. The phase of delocalized states is marginally stable. We demonstrate that the metal-insulator transition should be interpreted as a first-order phase transition. Consequences for perturbation approaches, the problem of self-averaging quantities and numerical scaling are discussed.