The phase diagram of the multi-dimensional Anderson localization via analytic determination of Lyapunov exponents

TL;DR

This paper analytically determines the phase diagram of Anderson localization in higher dimensions by calculating Lyapunov exponents, revealing different localization behaviors and a critical dimension at D=6.

Contribution

The authors generalize their analytical method to higher dimensions, providing exact calculations of Lyapunov exponents and characterizing the phase diagram of Anderson localization.

Findings

Existence of coexistence regions of extended and localized states for D>2.

Identification of a critical dimension D=6 for Anderson localization.

Distinction between exponential and non-exponential localization in different dimensions.

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.