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

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, $<{\psi }_{n,\mathbf{m}}^2>$, 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 ($2\le D\le 3$), where localized states are always exponentially localized and high-dimensional systems ($D\ge D_c=4$), 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.

arXiv comment17 pages, 5 figures, to appear in Eur. Phys.J.B

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• Reply to Comment on "Exact analytic solution for the generalized Lyapunov exponent of the 2-dimensional Anderson localization"

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• Random walk approach to the analytic solution of random systems with multiplicative noise - the Anderson localization problem
• Phase Diagram for Anderson Disorder: beyond Single-Parameter Scaling
• Analytical realization of finite-size scaling for Anderson localization. Does the band of critical states exist for d>2?