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

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 $⟨V^2⟩<\infty$ 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 $⟨V^2⟩<\infty$, one recovers Gaussian fluctuations with the variance equal to the mean value: ${\gamma }_2\simeq {\gamma }_1$. For $⟨V^2⟩=\infty$, 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 $g\to 0$.

6 pages, LaTeX

Cited by in corpus (4)

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