LOCALIZATION

Effects of the disorder (or free-orientation) on the multiple scattering situation by using discrete velocity models for the possible (dynamical) localization and/or delocalization of (plane) sound waves propagating in dilute monatomic hard-sphere gases are presented. After comparing with previous free-orientation (n = 2) results, we show that there also exists a certain gap of the spectra for the relevant periodic (wave-propagating) operator when the disorder or free-orientation exists and when a periodic medium with a gap (in spectra) is (slightly) randomized (like our orientation-free 6- and 8-velocity cases) then possible localization and/or delocalization occur in a vicinity of the edges of the gap, even when multiple scattering is being considered.

As a result of advances in experimental and theoretical physics, many interesting problems have arisen in condensed-matter physics, typically as a result of the quantum-mechanical nature of a system. Areas of interest include Anderson localization, universal conductance fluctuations, normal electron persistent currents, and the properties of quasicrystals. Understanding such systems is challenging because of complications arising from the large number of particles involved, intractable symmetries, the presence of time-dependent or nonlinear terms in the Schrödinger equation, etc. Some progress has been made by studying large scale classical analog experiments which may accurately model the salient quantum-mechanical features of a condensed-matter system. This paper describes research with a number of acoustical systems which have addressed contemporary problems in condensed-matter physics.

We discuss the localization behavior of localized electronic wave functions in the one- and two-dimensional tight-binding Anderson model with diagonal disorder. We find that the distributions of the local wave function amplitudes at fixed distances from the localization center are well approximated by log-normal fits which become exact at large distances. These fits are consistent with the standard single parameter scaling theory for the Anderson model in 1d, but they suggest that a second parameter is required to describe the scaling behavior of the amplitude fluctuations in 2d. From the log-normal distributions we calculate analytically the decay of the mean wave functions. For short distances from the localization center we find stretched exponential localization (sublocalization) in both, 1d and 2d. In 1d, for large distances, the mean wave functions depend on the number of configurations N used in the averaging procedure and decay faster that exponentially (superlocalization) converging to simple exponential behavior only in the asymptotic limit. In 2d, in contrast, the localization length increases logarithmically with the distance from the localization center and sublocalization occurs also in the second regime. The N-dependence of the mean wave functions is weak. The analytical result agrees remarkably well with the numerical calculations. (241kb)

States on a Sierpinski triangle are described using a formally exact and general Hamiltonian renormalization . The spectra of new (as well as previously examined) models are characterized . Numerical studies based on the renormalization suggest that the only models which exhibit absolutely continuous specta are effectively one-dimensional in the limit of large distances .

We present a method that allows us to find the asymptotic form of various characteristics of disordered systems in the strong localization regime, i.e., when either the random potential is big or the energy is close to a spectral edge. The method is based on the hypothesis that the relevant realizations of the random potential in the strong localization regime have the form of a collection of deep random wells that are uniformly and chaotically distributed in space with a sufficiently small density. Assuming this and using the density expansion, we show first that the density of wells coincides in leading order with the density of states. Thus the density of states is in fact the small parameter of the theory in the strong localization regime. Then we derive the Mott formula for the low frequency conductivity and the asymptotic formulae for certain two-point correlators when the difference of the respective energies is small.

The Anderson localization problem in one and two dimensions is solved analytically via the calculation of the generalized Lyapunov exponents. This is achieved by making use of signal theory. The phase diagram can be analysed in this way. In the one-dimensional case all states are localized for arbitrarily small disorder in agreement with existing theories. In the two-dimensional case for larger energies and large disorder all states are localized but for certain energies and small disorder extended and localized states coexist. The phase of delocalized states is marginally stable. We demonstrate that the metal–insulator transition should be interpreted as a first-order phase transition. Consequences for perturbation approaches, the problem of self-averaging quantities and numerical scaling are discussed.

The effects of disorder (or free-orientation) using the discrete velocity model for the possible (dynamical) localization of (plane) sound waves propagating in dilute monatomic hard-sphere gases are presented. Comparison with previous fixed-orientation ( [theta] = 0) results show that there exists a certain gap in the spectra when the disorder or free-orientation exists and when a periodic medium with a gap (in spectra) is (slightly) randomized (like our orientation-free 4-velocity case) possible localization occurs in the vicinity of the edges of the gap.

We found that the coherent backscattering cone of turbid samples containing spherical Mie scatterers with a size parameter larger than about 20 strongly deviates from known analytical curve shapes. We compare experimental data with numerical simulations of Monte Carlo type. Moreover, we present a new wide-angle coherent backscattering set-up.

The standard one-parameter scaling theory predicts that all eigenstates in two-dimensional random lattices are weakly localized. We show that this claim fails in two-dimensional dipolar Frenkel exciton systems. The linear energy dispersion at the top of the exciton band, originating from the long-range inter-site coupling of dipolar nature, yields the same size-scaling law for the level spacing and the effective disorder seen by the exciton. This finally results in the delocalization of those eigenstates in the thermodynamic limit. Large scale numerical simulations allow us to perform a detailed multifractal analysis and to elucidate the nature of the excitonic eigenstates.

Recently time-reversal techniques have emerged as a new, important and fascinating discipline within wave propagation. Many of the problems involved can best be understood, analysed and optimized based on a random field model for the medium. Here we discuss stable refocusing of second-order time-reversed reflections. This phenomenon may appear as surprising at first. However, we show how it can be understood in very simple terms viewing the wavefield as a stochastic process. We give sufficient conditions on Green's function of the propagation problem for the phenomenon to happen. In particular we discuss acoustic wave propagation in the regime of weak random medium fluctuations and explicitly give the derivation of stable refocusing in this case, illustrating it with numerical examples.

We report resistance measurements for a single multiwall carbon nanotube as a function of gate voltage and perpendicular magnetic field. The tubes were trapped onto pre-patterned Al electrodes by means of an ac electric field. Magnetoresistance traces measured for various values of the gate voltage were averaged, which corresponds to an ensemble averaging of conductance fluctuations induced by quantum interference. The ensemble averaging decreases the conductance fluctuations, while leaving the weak localization contribution to the resistance unchanged. Our data can be consistently interpreted in terms quantum transport in the presence of a weak disorder.

We consider the Friedel sum rule (FSR) in the context of the scattering theory for the Schrödinger operator -D _x

It is known that in high dimensions there exist abundant localized excitations such as dromions, lumps, ring soliton solutions and so on. In this paper, the possible chaotic and fractal localized structures are revealed for the (2+1)-dimensional dispersive long-wave equation. The chaotic and fractal dromion and lump patterns of the model are constructed by some types of lower-dimensional chaotic and fractal patterns.

In this Letter, we wish to discussion some basic questions pertinent to the phenomenon of Anderson localization of classical waves in two-dimensional random media. Although a definite answer to the two-dimensional localization is not yet found, a common consensus has been reached based upon the scaling analysis Abraham et al. [Phys. Rev. Lett. 43 (1979) 679]. That is, all waves are localized in two dimensions for any given amount of disorders. This view has been prevalent for more than two decades. Here, we explain some recent results and considerations that tend to be contrary to this view or the consequences of it.