Localization in one-dimensional chains with Lévy-type disorder

Abstract

We study Anderson localization of the classical lattice waves in a chain with mass impurities distributed randomly through a power-law relation $s^{-(1+\alpha )}$ with $s$ as the distance between two successive impurities and $\alpha >0$. This model of disorder is long-range correlated and is inspired by the peculiar structure of the complex optical systems known as Lévy glasses. Using theoretical arguments and numerics, we show that in the regime in which the average distance between impurities is finite with infinite variance, the small-frequency behaviour of the localization length is ${\xi }_{\alpha }(\omega )\sim {\omega }^{-\alpha }$. The physical interpretation of this result is that, for small frequencies and long wavelengths, the waves feel an effective disorder whose fluctuations are scale-dependent. Numerical simulations show that an initially localized wavepacket attains, at large times, a characteristic inverse power-law front with an $\alpha$-dependent exponent which can be estimated analytically.

References in corpus (18)

• Anomalous Localization in Low-Dimensional Systems with Correlated Disorder
• From anomalous energy diffusion to Levy walks and heat conductivity in one-dimensional systems
• Classical Representation of the 1D Anderson Model
• One-Dimensional Stochastic Lévy-Lorentz Gas
• Delocalization in harmonic chains with long-range correlated random masses
• Lévy walks and scaling in quenched disordered media
• Energy diffusion in hard-point systems
• Spreading in Disordered Lattices with Different Nonlinearities
• Nonalgebraic length dependence of transmission through a chain of barriers with a Levy spacing distribution
• Conductance through quantum wires with Levy-type disorder: universal statistics in anomalous quantum transport
• Transport and Scaling in Quenched 2D and 3D Lévy Quasicrystals
• Photonic heterostructures with Levy-type disorder: statistics of coherent transmission
• Asymptotic energy profile of a wavepacket in disordered chains
• Anomalous thermal properties of a harmonic chain with correlated isotopic disorder
• Anomalous transmission and drifts in one-dimensional Levy structures
• Lyapunov exponents in 1d disordered system with long-range memory
• Dynamics of a quantum particle in low-dimensional disordered systems with extended states
• Quantum Diffusion and Localization in Disordered Electronic Systems

Cited by in corpus (10)

• Superdiffusive Transport and Energy Localization in Disordered Granular Crystals
• Scattering of Waves by Impurities in Precompressed Granular Chains
• Dirac wave transmission in Lévy disordered systems
• Absence of Anderson localization in certain random lattices
• A Lloyd-model generalization: Conductance fluctuations in one-dimensional disordered systems
• Controlling Anderson localization in disordered heterostructures with Lévy-type distribution
• Localization properties of harmonic chains with correlated mass and spring disorder: Analytical approach
• Wave transmission and its universal fluctuations in one-dimensional systems with Lévy-like disorder: Schrödinger, Klein-Gordon and Dirac equations
• Localization of disordered harmonic chain with long-range correlation
• Localization in a one-dimensional alloy with an arbitrary distribution of spacing between impurities: Application to Lévy glass