Anomalies and non-log-normal tails in one-dimensional localization with power-law disorder

Abstract

Within a general framework, we discuss the wave function statistics in the Lloyd model of Anderson localization on a one-dimensional lattice with a Cauchy distribution for the random on-site potential. We demonstrate that already in leading order in the disorder strength, there exists a hierarchy of anomalies in the probability distributions of the wave function, the conductance, and the local density of states, for every energy which corresponds to a rational ratio of wave length to the lattice constant. We also show that these distribution functions do have power-law rather then log-normal tails and do not display universal single-parameter scaling. These peculiarities persist in any model with power-law tails of the disorder distribution function.

4 pages, 2 figures

References in corpus (6)

• Single parameter scaling in one-dimensional localization revisited
• Band-center anomaly of the conductance distribution in one-dimensional Anderson localization
• Single parameter scaling in 1-D Anderson localization. Exact analytical solution
• Statistics of finite-time Lyapunov exponents in a random time-dependent potential
• Scaling in the one-dimensional Anderson localization problem in the region of fluctuation states
• Microscopic versus mesoscopic local density of states in one-dimensional localization

Cited by in corpus (14)

• Anomalous Localization in Low-Dimensional Systems with Correlated Disorder
• Detangling Flat Bands into Fano Lattices
• Flat band states: disorder and nonlinearity
• Localization of weakly disordered flat band states
• Wigner time delay and related concepts -- Application to transport in coherent conductors
• Non-universality of Anderson localization in short-range correlated disorder
• Localizations on Complex Networks
• Large deviations of the Lyapunov exponent in 2D matrix Langevin dynamics with applications to one-dimensional Anderson Localization models
• Localization for one-dimensional random potentials with large local fluctuations
• Fluctuations of the product of random matrices and generalized Lyapunov exponent
• Crossover of conductance and local density of states in a single-channel disordered quantum wire
• A Lloyd-model generalization: Conductance fluctuations in one-dimensional disordered systems
• The generalized Lyapunov exponent for the one-dimensional Schrödinger equation with Cauchy disorder: some exact results
• Representation theory and products of random matrices in $\text{SL}(2,\mathbb{R})$