Single parameter scaling in 1-D Anderson localization. Exact analytical solution

Abstract

The variance of the Lyapunov exponent is calculated exactly in the one-dimensional Anderson model with random site energies distributed according to the Cauchy distribution. We derive an exact analytical criterion for the validity of the single parameter scaling in this model. According to this criterion, states with energies from the conduction band of the underlying non-random system satisfy single parameter scaling when disorder is small enough. At the same time, single parameter scaling is not valid for states close to band boundaries and those outside of the original spectrum even in the case of small disorder. The obtained results are applied to the Kronig-Penney model with the potential in the form of periodically positioned $\delta$-functions with random strengths. We show that an increase in the disorder can restore single parameter scalingbehavior for states from band-gaps of this model.

12 pages, 2 figures, RevTex, Submitted to PRB

Cited by in corpus (42)

• Anomalous Localization in Low-Dimensional Systems with Correlated Disorder
• Detangling Flat Bands into Fano Lattices
• Transport Phase Diagram and Anderson Localization in Hyperuniform Disordered Photonic Materials
• Localization of weakly disordered flat band states
• Wigner time delay and related concepts -- Application to transport in coherent conductors
• Band-center anomaly of the conductance distribution in one-dimensional Anderson localization
• Localization-delocalization transition in a one-dimensional system with long-range correlated off-diagonal disorder
• Scaling and the center of band anomaly in a one-dimensional Anderson model with diagonal disorder
• Fluctuations of the Lyapunov exponent in 2D disordered systems
• Signatures of photon localization
• Scaling Properties of 1D Anderson Model with Correlated Diagonal Disorder
• Anticorrelations from power-law spectral disorder and conditions for an Anderson transition
• Scaling in the one-dimensional Anderson localization problem in the region of fluctuation states
• Sublocalization, superlocalization, and violation of standard single parameter scaling in the Anderson model
• Anomalies and non-log-normal tails in one-dimensional localization with power-law disorder
• Localization properties of one-dimensional Frenkel excitons: Gaussian versus Lorentzian diagonal disorder
• Strong-disorder renormalization-group study of the one-dimensional tight-binding model
• Hybridization of wave functions in one-dimensional localization
• Perturbative test of single parameter scaling for 1D random media
• Anomalous localisation in the aperiodic Kronig-Penney model
• Localization for one-dimensional random potentials with large local fluctuations
• Fluctuations of the product of random matrices and generalized Lyapunov exponent
• Generalized Lyapunov Exponent and Transmission Statistics in One-dimensional Gaussian Correlated Potentials
• Statistics of transmission in one-dimensional disordered systems: universal characteristics of states in the fluctuation tails
• Near integrable systems
• Anomalous localization enhancement in one-dimensional non-Hermitian disordered lattices
• Crossover from diffusive to strongly localized regime in two-dimensional systems
• Transport properties of 1D disordered models: a novel approach
• A Lloyd-model generalization: Conductance fluctuations in one-dimensional disordered systems
• Conductance of tubular nanowires with disorder
• Unified description of perturbation theory and band center anomaly in one-dimensional Anderson localization
• Scaling properties of one-dimensional off-diagonal disorder
• Resistance of a 1D random chain: Hamiltonian version of the transfer matrix approach
• Anomalous scaling of conductance cumulants in one-dimensional Anderson localization
• Random-phase reservoir and a quantum resistor: The Lloyd model
• Resonant Anderson localization in segmented wires
• Scattering Expansion for Localization in One Dimension: from Disordered Wires to Quantum Walks
• The generalized Lyapunov exponent for the one-dimensional Schrödinger equation with Cauchy disorder: some exact results
• Statistical properties of a localization-delocalization transition induced by correlated disorder
• Scattering Expansion for Localization in One Dimension
• Exact transmission moments in one-dimensional weak localization and single-parameter scaling
• Wave localization in binary isotopically disordered one-dimensional harmonic chains with impurities having arbitrary cross section and concentration