Generalized Lyapunov exponent of random matrices and universality classes for SPS in 1D Anderson localisation

TL;DR

This paper investigates the generalized Lyapunov exponent for products of random matrices in 1D Anderson localization, revealing universal behaviors and different fluctuation regimes depending on the potential's distribution tail.

Contribution

It introduces a universal formula for the generalized Lyapunov exponent in different potential distribution regimes, connecting fluctuations to universality classes in 1D Anderson localization.

Findings

Universal formula for Lyapunov exponent in high energy/weak disorder limit

Gaussian fluctuations when potential has finite second moment

Non-Gaussian large deviations for heavy-tailed potentials

Abstract

Products of random matrix products of $\mathrm{SL}(2,\mathbb{R})$, corresponding to transfer matrices for the one-dimensional Schr\"odinger equation with a random potential $V$, are studied. I consider both the case where the potential has a finite second moment $\langle V^2\rangle<\infty$ and the case where its distribution presents a power law tail $p(V)\sim|V|^{-1-\alpha}$ for $0<\alpha<2$. I study the generalized Lyapunov exponent of the random matrix product (i.e. the cumulant generating function of the logarithm of the wave function). In the high energy/weak disorder limit, it is shown to be given by a universal formula controlled by a unique scale (single parameter scaling). For $\langle V^2\rangle<\infty$, one recovers Gaussian fluctuations with the variance equal to the mean value: $\gamma_2\simeq\gamma_1$. For $\langle V^2\rangle=\infty$, one finds $\gamma_2\simeq(2/\alpha)\,\gamma_1$ and non Gaussian large deviations, related to the universal limiting behaviour of the conductance distribution $W(g)\sim g^{-1+\alpha/2}$ for $g\to0$.