Localization in one-dimensional chains with Lévy-type disorder

Localization properties of harmonic chains with correlated mass and spring disorder: Analytical approach

We study the localization properties of normal modes in harmonic chains with mass and spring weak disorder. Using a perturbative approach, an expression for the localization length L_{loc} is obtained, which is valid for arbitrary correlations of the disorder (mass disorder correlations, spring disorder correlations, and mass-spring disorder correlations are allowed), and for practically the whole frequency band. In addition, we show how to generate effective mobility edges by the use of disorder with long range self-correlations and cross-correlations. The transport of phonons is also analyzed, showing effective transparent windows that can be manipulated through the disorder correlations even for relative short chain sizes. These results are connected to the problem of heat conduction in the harmonic chain; indeed, we discuss the size scaling of the thermal conductivity from the perturbative expression of L_{loc}. Our results may have applications in modulating thermal transport, particularly in the design of thermal filters or in manufacturing high-thermal-conductivity materials.

Wave transmission and its universal fluctuations in one-dimensional systems with Lévy-like disorder: Schrödinger, Klein-Gordon, and Dirac equations

We investigate the propagation of waves in one-dimensional systems with Lévy-type disorder. We perform a complete analysis of nonrelativistic and relativistic wave transmission submitted to potential barriers whose width, separation, or both follow Lévy distributions characterized by an exponent 0<α<1. For the first two cases, where one of the parameters is fixed, nonrelativistic and relativistic waves present anomalous localization, 〈T〉∝L^{-α}. However, for the latter case, in which both parameters follow a Lévy distribution, nonrelativistic and relativistic waves present a transition between anomalous and standard localization as the incidence energy increases relative to the barrier height. Moreover, we obtain the localization diagram delimiting anomalous and standard localization regimes, in terms of incidence angle and energy. Finally, we verify that transmission fluctuations, characterized by its standard deviation, are universal, independent of barrier architecture, wave equation type, incidence energy, and angle, further extending earlier studies on electronic localization.

Localization in a one-dimensional alloy with an arbitrary distribution of spacing between impurities: Application to Lévy glass

We have studied the localization of waves in a one-dimensional lattice consisting of impurities where the spacing between consecutive impurities can take certain values with given probabilities. In general, such a distribution of impurities induces correlations in the disorder. In particular, with a power-law distribution of spacing, this system is used as a model for light propagation in Lévy glasses. We introduce a method of calculating the Lyapunov exponent which overcomes limitations in the previous studies and can be easily extended to higher orders of perturbation theory. We obtain the Lyapunov exponent up to fourth order of perturbation and discuss the range of validity of perturbation theory, transparent states, and anomalous energies which are characterized by divergences in different orders of the expansion. We also carry out numerical simulations which are in agreement with our analytical results.

Heat conduction in harmonic chains with Lévy-type disorder

We consider heat transport in a one-dimensional harmonic chain attached at its ends to Langevin heat baths. The harmonic chain has mass impurities where the separation d between any two successive impurities is randomly distributed according to a power-law distribution P(d)∼1/d^{α+1}, being α>0. In the regime where the first moment of the distribution is well defined (1<α<2) the thermal conductivity κ scales with the system size N as κ∼N^{(α-3)/α} for fixed boundary conditions, whereas for free boundary conditions κ∼N^{(α-1)/α} if N≫1. When α=2, the inverse localization length λ scales with the frequency ω as λ∼ω^{2}lnω in the low-frequency regime, due to the logarithmic correction, the size scaling law of the thermal conductivity acquires a nonclosed form. When α>2, the thermal conductivity scales as in the uncorrelated disorder case. The situation α<1 is only analyzed numerically, where λ(ω)∼ω^{2-α}, which leads to the following asymptotic thermal conductivity: κ∼N^{-(α+1)/(2-α)} for fixed boundary conditions and κ∼N^{(1-α)/(2-α)} for free boundary conditions.

Dirac wave transmission in Lévy-disordered systems

We investigate the propagation of electronic waves described by the Dirac equation subject to a Lévy-type disorder distribution. Our numerical calculations, based on the transfer matrix method, in a system with a distribution of potential barriers show that it presents a phase transition from anomalous to standard to anomalous localization as the incidence energy increases. In contrast, electronic waves described by the Schrödinger equation do not present such transitions. Moreover, we obtain the phase diagram delimiting anomalous and standard localization regimes, in the form of an incidence angle versus incidence energy diagram, and argue that transitions can also be characterized by the behavior of the dispersion of the transmission. We attribute this transition to an abrupt reduction in the transmittance of the system when the incidence angle is higher than a critical value which induces a decrease in the transmission fluctuations.

Information-Length Scaling in a Generalized One-Dimensional Lloyd's Model

We perform a detailed numerical study of the localization properties of the eigenfunctions of one-dimensional (1D) tight-binding wires with on-site disorder characterized by long-tailed distributions: For large ϵ , P ( ϵ ) ∼ 1 / ϵ 1 + α with α ∈ ( 0 , 2 ] ; where ϵ are the on-site random energies. Our model serves as a generalization of 1D Lloyd's model, which corresponds to α = 1 . In particular, we demonstrate that the information length β of the eigenfunctions follows the scaling law β = γ x / ( 1 + γ x ) , with x = ξ / L and γ ≡ γ ( α ) . Here, ξ is the eigenfunction localization length (that we extract from the scaling of Landauer's conductance) and L is the wire length. We also report that for α = 2 the properties of the 1D Anderson model are effectively reproduced.

Absence of Anderson localization in certain random lattices

We report on the transition between an Anderson localized regime and a conductive regime in a one-dimensional microwave scattering system with correlated disorder. We show experimentally that when long-range correlations are introduced, in the form of a power-law spectral density with power larger than 2, the localization length becomes much bigger than the sample size and the transmission peaks typical of an Anderson localized system merge into a pass band. As other forms of long-range correlations are known to have the opposite effect, i.e., to enhance localization, our results show that care is needed when discussing the effects of correlations, as different kinds of long-range correlations can give rise to very different behavior.

Scattering of waves by impurities in precompressed granular chains

We study scattering of waves by impurities in strongly precompressed granular chains. We explore the linear scattering of plane waves and identify a closed-form expression for the reflection and transmission coefficients for the scattering of the waves from both a single impurity and a double impurity. For single-impurity chains, we show that, within the transmission band of the host granular chain, high-frequency waves are strongly attenuated (such that the transmission coefficient vanishes as the wavenumber k→±π), whereas low-frequency waves are well-transmitted through the impurity. For double-impurity chains, we identify a resonance-enabling full transmission at a particular frequency-in a manner that is analogous to the Ramsauer-Townsend (RT) resonance from quantum physics. We also demonstrate that one can tune the frequency of the RT resonance to any value in the pass band of the host chain. We corroborate our theoretical predictions both numerically and experimentally, and we directly observe almost complete transmission for frequencies close to the RT resonance frequency. Finally, we show how this RT resonance can lead to the existence of reflectionless modes in granular chains (including disordered ones) with multiple double impurities.

Lloyd-model generalization: Conductance fluctuations in one-dimensional disordered systems

We perform a detailed numerical study of the conductance G through one-dimensional (1D) tight-binding wires with on-site disorder. The random configurations of the on-site energies ε of the tight-binding Hamiltonian are characterized by long-tailed distributions: For large ε, P(ε)∼1/ε^{1+α} with α∈(0,2). Our model serves as a generalization of the 1D Lloyd model, which corresponds to α=1. First, we verify that the ensemble average 〈-lnG〉 is proportional to the length of the wire L for all values of α, providing the localization length ξ from 〈-lnG〉=2L/ξ. Then, we show that the probability distribution function P(G) is fully determined by the exponent α and 〈-lnG〉. In contrast to 1D wires with standard white-noise disorder, our wire model exhibits bimodal distributions of the conductance with peaks at G=0 and 1. In addition, we show that P(lnG) is proportional to G^{β}, for G→0, with β≤α/2, in agreement with previous studies.

Superdiffusive transport and energy localization in disordered granular crystals

We study the spreading of initially localized excitations in one-dimensional disordered granular crystals. We thereby investigate localization phenomena in strongly nonlinear systems, which we demonstrate to differ fundamentally from localization in linear and weakly nonlinear systems. We conduct a thorough comparison of wave dynamics in chains with three different types of disorder-an uncorrelated (Anderson-like) disorder and two types of correlated disorders (which are produced by random dimer arrangements)-and for two types of initial conditions (displacement excitations and velocity excitations). We find for strongly precompressed (i.e., weakly nonlinear) chains that the dynamics depend strongly on the type of initial condition. In particular, for displacement excitations, the long-time asymptotic behavior of the second moment m̃(2) of the energy has oscillations that depend on the type of disorder, with a complex trend that differs markedly from a power law and which is particularly evident for an Anderson-like disorder. By contrast, for velocity excitations, we find that a standard scaling m̃(2)∼t(γ) (for some constant γ) applies for all three types of disorder. For weakly precompressed (i.e., strongly nonlinear) chains, m̃(2) and the inverse participation ratio P(-1) satisfy scaling relations m̃(2)∼t(γ) and P(-1)∼t(-η), and the dynamics is superdiffusive for all of the cases that we consider. Additionally, when precompression is strong, the inverse participation ratio decreases slowly (with η<0.1) for all three types of disorder, and the dynamics leads to a partial localization around the core and the leading edge of a propagating wave packet. For an Anderson-like disorder, displacement perturbations lead to localization of energy primarily in the core, and velocity perturbations cause the energy to be divided between the core and the leading edge. This localization phenomenon does not occur in the sonic-vacuum regime, which yields the surprising result that the energy is no longer contained in strongly nonlinear waves but instead is spread across many sites. In this regime, the exponents are very similar (roughly γ≈1.7 and η≈1) for all three types of disorder and for both types of initial conditions.