Perturbative test of single parameter scaling for 1D random media

TL;DR

This paper investigates the validity of single parameter scaling in 1D random media, demonstrating it holds at lowest order but is violated at higher orders, with explicit examples from the Anderson model.

Contribution

It provides a perturbative proof of single parameter scaling validity at lowest order and clarifies its limitations at higher orders in 1D random media.

Findings

Single parameter scaling holds at lowest order in disorder strength.

Higher order effects lead to violations of single parameter scaling.

Explicit example provided for the Anderson model.

Abstract

Products of random matrices associated to one-dimensional random media satisfy a central limit theorem assuring convergence to a gaussian centered at the Lyapunov exponent. The hypothesis of single parameter scaling states that its variance is equal to the Lyapunov exponent. We settle discussions about its validity for a wide class of models by proving that, away from anomalies, single parameter scaling holds to lowest order perturbation theory in the disorder strength. However, it is generically violated at higher order. This is explicitely exhibited for the Anderson model.