Scaling Properties in Highly Anisotropic Systems

TL;DR

This paper extends the scaling theory of conductances and localization lengths to anisotropic systems, revealing relationships between localization lengths, conductivities, and anisotropy strength in two-dimensional systems.

Contribution

It generalizes scaling functions to anisotropic systems and establishes quantitative relationships between localization lengths and conductivities based on anisotropy.

Findings

Scaling functions are recovered when system dimensions are proportional to localization lengths.

The geometric mean of localization lengths depends on the geometric mean of conductivities.

The ratio of localization lengths scales with the square root of the conductivity ratio, proportional to anisotropy strength t.

Abstract

Scaling of the conductances and the finite-size localization lengths is generalized to anisotropic systems and tested in two dimensional systems. Scaling functions of isotropic systems are recovered once the dimension of the system in each direction is chosen to be proportional to the localization length. It is also shown that the geometric mean of the localization lengths is a function of the geometric mean of the conductivities. The ratio of the localization lengths is proportional to the square root of the ratio of the conductivities, which in turn is proportional to the anisotropy strength t, in the weak scattering limit.