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

Christophe Texier

doi:10.1209/0295-5075/131/17002

All issues

Volume 131 / No 1 (July 2020)

EPL, 131 1 (2020) 17002

Abstract

Issue		EPL Volume 131, Number 1, July 2020


Article Number		17002
Number of page(s)		6
Section		Condensed Matter: Electronic Structure, Electrical, Magnetic and Optical Properties
DOI		https://doi.org/10.1209/0295-5075/131/17002
Published online		29 July 2020

EPL, 131 (2020) 17002

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

Christophe Texier

LPTMS, CNRS, Université Paris-Saclay - 91405 Orsay Cedex, France

Received: 5 May 2020
Accepted: 24 June 2020

Abstract

Products of random matrix products of $\mathrm{SL}(2,\mathbb{R})$ , corresponding to transfer matrices for the one-dimensional Schrödinger 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 $g\to0$ .

PACS: 73.20.Fz – Weak or Anderson localization / 02.10.Yn – Matrix theory / 02.50.-r – Probability theory, stochastic processes, and statistics

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