Analytical realization of finite-size scaling for Anderson localization: Is there a transition in the 2D case?

Roughly half the numerical investigations of the Anderson transition are based on consideration of an associated quasi-1D system and postulation of one-parameter scaling for the minimal Lyapunov exponent. If this algorithm is taken seriously, it leads to unambiguous prediction of the 2D phase transition. The transition is of the Kosterlitz-Thouless type and occurs between exponential and power law localization (Pichard and Sarma, 1981). This conclusion does not contradict numerical results if raw data are considered. As for interpretation of these data in terms of one-parameter scaling, this is inadmissible: the minimal Lyapunov exponent does not obey any scaling. A scaling relation is valid not for a minimal, but for some effective Lyapunov exponent whose dependence on the parameters is determined by the scaling itself. If finite-sizedd scaling is based on the effective Lyapunov exponent, the existence of the 2D transition becomes indefinite, but still rather probable. Interpretation of the results in terms of the Gell-Mann-Low equation is also given.