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

Abstract

The Anderson localization problem in one and two dimensions is solvedanalytically via the calculation of the generalized Lyapunov exponents. Thisis achieved by making use of signal theory. The phase diagram can beanalysed in this way. In the one-dimensional case all states are localizedfor arbitrarily small disorder in agreement with existing theories. In thetwo-dimensional case for larger energies and large disorder all states arelocalized but for certain energies and small disorder extended and localizedstates coexist. The phase of delocalized states is marginally stable. Wedemonstrate that the metal–insulator transition should be interpreted as afirst-order phase transition. Consequences for perturbation approaches, theproblem of self-averaging quantities and numerical scaling are discussed.