Localization and Mobility Edge in One-Dimensional Potentials with Correlated Disorder

We show that a mobility edge exists in 1D random potentials provided specific long-range correlations. Our approach is based on the relation between binary correlator of a site potential and the localization length. We give the algorithm to construct numerically potentials with mobility edge at any given energy inside allowed zone. Another natural way to generate such potentials is to use chaotic trajectories of non-linear maps. Our numerical calculations for few particular potentials demonstrate the presence of mobility edges in 1D geometry.Comment: 4 pages in RevTex and 2 Postscript figures; revised version published in Phys. Rev. Lett. 82 (1999) 406

The scaling limit of the critical one-dimensional random Schrodinger operator

We consider two models of one-dimensional discrete random Schrodinger operators (H_n \psi)_l ={\psi}_{l-1}+{\psi}_{l +1}+v_l {\psi}_l, {\psi}_0={\psi}_{n+1}=0 in the cases v_k=\sigma {\omega}_k/\sqrt{n} and v_k=\sigma {\omega}_k/ \sqrt{k}. Here {\omega}_k are independent random variables with mean 0 and variance 1. We show that the eigenvectors are delocalized and the transfer matrix evolution has a scaling limit given by a stochastic differential equation. In both cases, eigenvalues near a fixed bulk energy E have a point process limit. We give bounds on the eigenvalue repulsion, large gap probability, identify the limiting intensity and provide a central limit theorem. In the second model, the limiting processes are the same as the point processes obtained as the bulk scaling limits of the beta-ensembles of random matrix theory. In the first model, the eigenvalue repulsion is much stronger.Comment: 36 pages, 2 figure

Wave transmission, phonon localization and heat conduction of 1D Frenkel-Kontorova chain

We study the transmission coefficient of a plane wave through a 1D finite quasi-periodic system -- the Frenkel-Kontorova (FK) model -- embedding in an infinite uniform harmonic chain. By varying the mass of atoms in the infinite uniform chain, we obtain the transmission coefficients for {\it all} eigenfrequencies. The phonon localization of the incommensurated FK chain is also studied in terms of the transmission coefficients and the Thouless exponents. Moreover, the heat conduction of Rubin-Greer-like model for FK chain at low temperature is calculated. It is found that the stationary heat flux $J(N)\sim N^{\alpha}$, and $\alpha$ depends on the strength of the external potential.Comment: 15 pages in Revtex, 8 EPS figure

Excitation of Small Quantum Systems by High-Frequency Fields

The excitation by a high frequency field of multi--level quantum systems with a slowly varying density of states is investigated. A general approach to study such systems is presented. The Floquet eigenstates are characterized on several energy scales. On a small scale, sharp universal quasi--resonances are found, whose shape is independent of the field parameters and the details of the system. On a larger scale an effective tight--binding equation is constructed for the amplitudes of these quasi--resonances. This equation is non--universal; two classes of examples are discussed in detail.Comment: 4 pages, revtex, no figure

Force Distribution in a Granular Medium

We report on systematic measurements of the distribution of normal forces exerted by granular material under uniaxial compression onto the interior surfaces of a confining vessel. Our experiments on three-dimensional, random packings of monodisperse glass beads show that this distribution is nearly uniform for forces below the mean force and decays exponentially for forces greater than the mean. The shape of the distribution and the value of the exponential decay constant are unaffected by changes in the system preparation history or in the boundary conditions. An empirical functional form for the distribution is proposed that provides an excellent fit over the whole force range measured and is also consistent with recent computer simulation data.Comment: 6 pages. For more information, see http://mrsec.uchicago.edu/granula

Diffusion in disordered systems under iterative measurement

We consider a sequence of idealized measurements of time-separation $\Delta t$ onto a discrete one-dimensional disordered system. A connection with Markov chains is found. For a rapid sequence of measurements, a diffusive regime occurs and the diffusion coefficient $D$ is analytically calculated. In a general point of view, this result suggests the possibility to break the Anderson localization due to decoherence effects. Quantum Zeno effect emerges because the diffusion coefficient $D$ vanishes at the limit $\Delta t \to 0$.Comment: 8 pages, 0 figures, LATEX. accepted in Phys.Rev.

Delocalization in Continuous Disordered Systems

Continuous One-dimensional models supporting extended states are studied. These delocalized statesoccur at well defined values of the energy and are consequences of simple statistical correlation rules. We explicitly study alloys of delta-barrier potentials as well as alloys and liquids of quantum well as.The divergence of the localization length is studied and a critical exponent 2/3 is found for the delta-barrier case, whereas for the quantum wells we find an exponent of 2 or 2/3 depending on the well's parameters. These results support the idea that correlations between random scattering sequences break Anderson localization. We further calculate the conductance of disordered superlattices. At the peak transmission the relative fluctuations of the transmission coefficient are vanishing.Comment: 8 page