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

The method proposed by the present authors to deal analytically with the problem of Anderson localization via disorder [J. Phys.: Condens. Matter 14, 13777 (2002)] is generalized for higher spatial dimensions D. In this way the generalized Lyapunov exponents for diagonal correlators of the wave function, 〈ψ_n,m²〉, 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 ≥ D_c = 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.