Volume 131, Number 1, July 2020
|Number of page(s)||6|
|Section||Condensed Matter: Electronic Structure, Electrical, Magnetic and Optical Properties|
|Published online||29 July 2020|
Generalized Lyapunov exponent of random matrices and universality classes for SPS in 1D Anderson localisation
LPTMS, CNRS, Université Paris-Saclay - 91405 Orsay Cedex, France
Received: 5 May 2020
Accepted: 24 June 2020
Products of random matrix products of , 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 and the case where its distribution presents a power law tail for . 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 , one recovers Gaussian fluctuations with the variance equal to the mean value: . For , one finds and non-Gaussian large deviations, related to the universal limiting behaviour of the conductance distribution for .
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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