Exact analytic solution of the multi-dimensional Anderson localization

TL;DR

This paper provides an exact analytical solution for Anderson localization in higher dimensions, revealing phase diagrams, coexistence of states, and critical dimensions, thus advancing understanding of localization phenomena.

Contribution

The authors generalize their analytical method to higher dimensions, enabling exact calculation of Lyapunov exponents and detailed phase diagrams for Anderson localization.

Findings

Existence of energy and disorder intervals with coexisting extended and localized states.

Identification of a first-order metal-insulator transition in dimensions greater than two.

Determination of the upper critical dimension as D=6 for Anderson localization.

Abstract

The method proposed by the present authors to deal analytically with the problem of Anderson localization via disorder [J.Phys.: Condens. Matter {\bf 14} (2002) 13777] is generalized for higher spatial dimensions D. In this way the generalized Lyapunov exponents for diagonal correlators of the wave function, $<\psi^2_{n,\mathbf{m}}>$, can be calculated analytically and exactly. This permits to determine the phase diagram of the system. For all dimensions $D > 2$ one finds intervals in the energy and the disorder where extended and localized states coexist: the metal-insulator transition should thus be interpreted as a first-order transition. The qualitative differences permit to group the systems into two classes: low-dimensional systems ($2\leq D \leq 3$), where localized states are always exponentially localized and high-dimensional systems ($D\geq D_c=4$), where states with non-exponential localization are also formed. The value of the upper critical dimension is found to be $D_0=6$ for the Anderson localization problem; this value is also characteristic of a related problem - percolation. Consequences for numerical scaling and other approaches are discussed in detail.