Two-Parameter Scaling Law of the Anderson Transition

TL;DR

This paper demonstrates that the Anderson transition in 3D follows a two-parameter scaling law, with critical exponents and a scaling function derived from high-precision numerical studies, revealing new insights into localization phenomena.

Contribution

The study introduces a two-parameter scaling law for the Anderson transition in 3D and provides detailed numerical analysis of the critical exponents and scaling functions.

Findings

Two-parameter scaling law for 3D Anderson transition

Critical exponents and scaling functions calculated

Signatures of Anderson transition may appear in 2D strongly localized regime

Abstract

It is shown that the Anderson transition (AT) in 3d obeys a two-parameter scaling law, derived from a pair of anisotropic scaling transformations, and corresponding critical exponents and scaling function calculated, using a high-precision numerical finite-size scaling study of the smallest Lyapunov exponent of quasi-1d systems of rectangular cross-section of L times l atoms in the limit of infinite L and l < L, for x=l/L ranging from 1/30 to 1/4. The second parameter is x, and there are two singularities: apart from the two-parameter scaling describing AT for x>0, corrections to scaling due to the irrelevant scaling field diverge when x->0, and the corresponding crossover length scale is also estimated. Furthermore, results suggest that the signatures of the AT in 3d should be present also in 2d strongly localized regime.