Exact analytic solution of the multi-dimensional Anderson localization

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The method proposed by the present authors to deal analytically with the problem of Anderson localization via disorder [J.Phys.: Condens. Matter 14 (2002) 13777] is generalized for higher spatial dimensions D. In this way the generalized Lyapunov exponents for diagonal correlators of the wave function, <ψ²ₙ,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≤ D ≤ 3), where localized states are always exponentially localized and high-dimensional systems (D≥ Dc=4), where states with non-exponential localization are also formed. The value of the upper critical dimension is found to be D₀=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.