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

TL;DR

This paper critically examines the finite-size scaling approach in Anderson localization, questioning the existence of a 2D phase transition and proposing that an effective Lyapunov exponent, rather than the minimal one, should be used for scaling analysis.

Contribution

It provides an analytical framework for finite-size scaling in Anderson localization and challenges the conventional interpretation of the 2D transition based on the minimal Lyapunov exponent.

Findings

The 2D Anderson transition is of the Kosterlitz-Thouless type.

The minimal Lyapunov exponent does not obey one-parameter scaling.

An effective Lyapunov exponent may reveal a probable 2D transition.

Abstract

Roughly half of 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 unumbiguous 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 the raw data are considered. As for interpretation of these data in terms of one-parameter scaling, such interpretation is inadmissible: the minimal Lyapunov exponent does not obey any scaling. A scaling relation is valid not for minimal, but for some effective Lyapunov exponent, whose dependence on parameters is determined by scaling itself. If finite-size scaling is based on the effective Lyapunov exponent, existence of the 2D transition becomes not definite, but still rather probable. Interpretation of the results in terms of the Gell-Mann -- Low equation is also given.