Narrow Results

We give a short summary on how to combine and extend results of Combes and Hislop [2] (short range Anderson model with additional displacements), Kirsch, Stollmann and Stolz [13] and [14] (long range Anderson model without displacements) to get localization in an energy interval above the infimum of the almost sure spectrum for a continuous multidimensional Anderson model including long range potentials and displacements.

As a supplement of our previous work [10], we consider the localized region of the random Schrödinger operators on l²(Z^d) and study the point process composed of their eigenvalues and corresponding localization centers. For the Anderson model we show that, this point process in the natural scaling limit converges in distribution to the Poisson process on the product space of energy and space. In other models with suitable Wegner-type bounds, we can at least show that limiting point processes are infinitely divisible.

We study a multi-particle quantum graph with random potential. Taking the approach of multiscale analysis, we prove exponential and strong dynamical localization of any order in the Hilbert–Schmidt norm near the spectral edge. Apart from the results on multi-particle systems, we also prove Lifshitz-type asymptotics for single-particle systems. This shows in particular that localization for single-particle quantum graphs holds under a weaker assumption on the random potential than previously known.

In this note, we review some results on localization and related properties for random Schrödinger operators arising in aperiodic media. These include the Anderson model associated to disordered quasicrystals and also the so-called Delone operators, operators associated to deterministic aperiodic structures.

It is the purpose of the present article to show that so-called network models, originally designed to describe static properties of disordered electronic systems, can be easily generalized to quantum-dynamical models, which then allow for an investigation of dynamical and spectral aspects. This concept is exemplified by the Chalker–Coddington model for the quantum Hall effect and a three-dimensional generalization of it. We simulate phase coherent diffusion of wave packets and consider spatial and spectral correlations of network eigenstates as well as the distribution of (quasi-)energy levels. Apart from that, it is demonstrated how network models can be used to determine two-point conductances. Our numerical calculations for the three-dimensional model at the Metal-Insulator transition point delivers, among others, an anomalous diffusion exponent of η=3-D₂=1.7±0.1. The methods presented here in detail have been used partially in earlier work.

We review the present status of the Anderson transition in the spectrum of the Dirac operator of QCD-like theories on the lattice. Localized modes at the low end of the spectrum have been found in SU(2) Yang–Mills theory with overlap and staggered valence fermions as well as in N_f = 2+1 QCD with staggered quarks. We draw an analogy between the transition from localized to delocalized modes in the Dirac spectrum and the Anderson transition in electronic systems. The QCD transition turns out to be in the same universality class as the transition in the corresponding Anderson model. We also speculate on the possible physical relevance of this transition to QCD at high temperature and the possible finite temperature phase transition in QCD-like models with different fermion contents.

The integer quantum Hall effect is studied for a non-interacting electron in a monolayer graphene. We numerically calculate the Hall conductivity in a single Landau level with disorder, and estimate the critical energies for the extended states in thermodynamic limit. We show that a valley-degenerated (K and K′) Landau band has extended levels at two different energies, indicating that an extra Hall plateau appears inside.

A Kronig–Penney model with short-range correlated impurities is used to investigate the electronic transport properties of one-dimensional disordered systems simulating both superlattices, protein chains, and polymers. Physical magnitudes characterizing the electronic charge transfer are analytically the transmission coefficient and the localization length. It is shown that a set of delocalized states occurs at particular well-defined energies. The divergence of the localization length is investigated and a critical exponent 1 is found for the barrier case which differs with the results of literature. These results support the idea that the nature of the extended states depend on the typical nature of the correlation in disorder. On this basis, the conductance of disordered superlattices is investigated as well as the relative fluctuations of the transmission coefficient.

Periodicity (in time or space) is a part and parcel of every living being: one can see, hear and feel it. Everyday examples are locomotion, respiration and heart beat. The reinforced N-dimensional periodicity over two or more crystalline solids results in the so-called phononic band gap crystals. These can have dramatic consequences on the propagation of phonons, vibrations and sound. The fundamental physics of cleverly fabricated phononic crystals can offer a systematic route to realize the Anderson localization of sound and vibrations. As to the applications, the phononic crystals are envisaged to find ways in the architecture, acoustic waveguides, designing transducers, elastic/acoustic filters, noise control, ultrasonics, medical imaging and acoustic cloaking, to mention a few. This review focuses on the brief sketch of the progress made in the field that seems to have prospered even more than was originally imagined in the early nineties.

A variety of heterogeneous potentials are capable of localizing linear noninteracting waves. In this work, we review different examples of heterogeneous localizing potentials which were realized in experiments. We then discuss the impact of nonlinearity induced by wave interactions, in particular, its destructive effect on the localizing properties of the heterogeneous potentials.

We theoretically investigate the optical Anderson localization in the disordered photorefractive lattices under different strength of saturable self-defocusing nonlinearity. Via continuously increasing the value of the nonlinear parameter and applying numerical simulations respectively, a critical behavior, a transformation from a localized state to a spreading state, is found. We use different initial power of the probe and obtain a series of critical values of the nonlinear parameter. Below the critical value, the localizing effect is enhanced with the increase of the nonlinear parameter. While above the critical value, the localized state is destroyed. We give a study of the critical behavior and the critical values corresponding to different physical parameters, including the intensity of the self-defocusing nonlinearity, the intensity of the lattice-forming beams and the randomness strength of the structure.

We revisit the problem of Anderson localization in a trapped Bose–Einstein condensate in 1D and 3D in a disordered potential, applying Quantum Monte Carlo technique because the disorder cannot be treated accurately in a perturbative way as even a small amount of disorder can produce dramatic changes in the physical properties of the system under investigation. Till date no unambiguous evidence of localization has been observed for matter waves in 3D. Matter waves made up of cold atoms are good candidates for such investigations. Simulations are performed for Rb gas in continuous space using canonical ensemble in the case of random and quasi-periodic potentials. To realize random and quasiperiodic potentials numerically we use speckle and bichromatic potentials, respectively. Owing to the high degree of control over the system parameters we specifically study the interplay of disorder and interaction in the system. A dilute Bose gas placed in a random environment falls into a fragmented localized state and the ergodicity (the repetitiveness of the wave function) is lost. An arbitrary Interaction can slowly overcome the effect of disorder and restore the ergodicity again. We observe that as the interaction strength increases, the wave functions become more and more delocalized. Since vanishing of Lyapunov exponent is only a necessary but not a sufficient condition for delocalization for probing the localization we calculate the mean square displacements as an alternative measure of localization. The path integral Monte Carlo technique in this paper numerically establishes the existing predictions of the scaling theory so far and paves a clear path for the further investigation of scaling theory to calculate more complicated properties like ‘critical exponents’ etc. in disordered quantum gases.

We report a numerical analysis of Anderson localization in a model of a doped semiconductor. The model incorporates the disorder arising from the random spatial distribution of the donor impurities and takes account of the electron-electron interactions between the carriers using density functional theory in the local density approximation. Preliminary results suggest that the model exhibits a metal-insulator transition.

We employed a first-principles theory – the supersymmetric field theory – formulated for wave transport in very general open media to study static transport of waves in quasi-one-dimensional localized samples. We predicted analytically and confirmed numerically that in these systems, localized waves display an unconventional diffusive phenomenon. Different from the prevailing self-consistent local diffusion model, our theory is capable of capturing all disorder-induced resonant transmissions, which give rise to significant enhancement of local diffusion inside a localized sample. Our theory should be able to be generalized to two- and three-dimensional open media, and open a new direction in the study of Anderson localization in open media.

At the Anderson metal-insulator transition the eigenstates develop multifractal fluctuations. Therefore their properties are intermediate between being extended and localized. As a result these wave functions are power-law correlated, which causes a substantial suppression of the local density of states at some random positions, resembling random local pseudogaps at the Fermi energy. Consequently the Kondo screening of magnetic moments is suppressed when a magnetic impurity happens to be at such a position. Due to these unscreened magnetic moments the critical exponents and multifractal dimensions at the metal-insulator transition take their smaller, unitary ensemble values for exchange couplings not exceeding a certain critical value J* ≈ .3D, where D is the band width. Here we present numerical calculations of the distribution of Kondo temperatures for the critical Power-law Band Random Matrix (PBRM) ensemble, whose properties are similar to that of the Anderson transition with the advantage of using a continuous parameter for tuning the generalized multifractal dimensions of the eigenstates.

Field transmission coefficients for microwave radiation between arrays of points on the incident and output surfaces of random samples are analyzed to yield the underlying quasi-normal modes and transmission eigenchannels of each realization of the sample. The linewidths, central frequencies, and transmitted speckle patterns associated with each of the modes of the medium are found. Modal speckle patterns are found to be strongly correlated leading to destructive interference between modes. This explains distinctive features of transmission spectra and pulsed transmission. An alternate description of wave transport is obtained from the eigenchannels and eigenvalues of the transmission matrix. The maximum transmission eigenvalue, τ₁ is near unity for diffusive waves even in turbid samples. For localized waves, τ₁ is nearly equal to the dimensionless conductance, which is the sum of all transmission eigenvalues, g = Στ_n. The spacings between the ensemble averages of successive values of lnτ_n are constant and equal to the inverse of the bare conductance in accord with predictions by Dorokhov. The effective number of transmission eigenvalues N_eff determines the contrast between the peak and background of radiation focused for maximum peak intensity. The connection between the mode and channel approaches is discussed.

Network models are both simple and general. In solid states physics, quantum network models that take into account the wave nature of the electron have been the subject of intense research. One example is the Chalker–Coddington model that describes the quantum Hall effect (J. T. Chalker, P. D. Coddington, J. Phys., C 21 2665 (1988)). In this paper we focus on the transfer matrices that describe these models. We emphasize the symmetry properties of these matrices and the consequences for eigenvalue and transport properties.

In this paper we deal with the general subject of realizing disordered states in optical lattices by using an unequal mixture of fast and slow (or frozen) particles. We discuss the onset of Anderson localization of fast hardcore bosons when brought into interaction with the random potential created by secondary hardcore bosons frozen in a superfluid state. In the case of softcore bosons we discuss how localization phenomena, in the form of fragmentation of the mixture into many metastable droplets, intervene when trying to reach the equilibrium ground state of the system.

In 1958, P.W. Anderson predicted the exponential localization¹ of electronic wave functions in disordered crystals and the resulting absence of diffusion. It was realized later that Anderson localization (AL) is ubiquitous in wave physics² as it originates from the interference between multiple scattering paths, and this has prompted an intense activity. Experimentally, localization has been reported in light waves³ microwaves,⁴ sound waves,⁵ and electron gases⁶ but to our knowledge there was no direct observation of exponential spatial localization of matter-waves (electrons or others). We present in this proceeding the experiment that lead to the observation of Anderson Localization (AL)⁷ of a Bose-Einstein Condensate (BEC) released into a one-dimensional waveguide in the presence of a controlled disorder created by laser speckle. Direct imaging allows for unambiguous observation of an exponential decay of the wavefunction when the conditions for AL are fulfilled. The disorder is created with a one-dimensional speckle potential whose noise spectrum has a high spatial frequency cut-off, hindering the observation of exponential localization if the expanding BEC contains atomic de Broglie wavelengths that are smaller than an effective mobility edge corresponding to that cut-off. In this case, we observe the density profiles that decay algebraically.⁹

One of the most intriguing phenomena in physics is the localization of waves in disordered media. This phenomenon was originally predicted by P. W. Anderson, fifty years ago, in the context of transport of electrons in crystals, but it has never been directly observed for matter waves. Ultracold atoms open a new scenario for the study of disorder-induced localization, due to the high degree of control of most of the system parameters, including interactions. For the first time we have employed a noninteracting Bose-Einstein condensate to study Anderson localization. The experiment is performed in a 1D lattice with quasi-periodic disorder, a system which features a crossover between extended and exponentially localized states as in the case of purely random disorder in higher dimensions. We clearly demonstrate localization by investigating transport properties, spatial and momentum distributions. Since the interaction in the condensate can be controlled, this system represents a novel tool to solve fundamental questions on the interplay of disorder and interactions and to explore exotic quantum phases.