Quantum Mesoscopic Scattering: Disordered Systems and Dyson Circular Ensembles

We consider elastic reflection and transmission of electrons by a disordered system characterized by a $2N\!\times\!2N$ scattering matrix $S$. Expressing $S$ in terms of the $N$ radial parameters and of the four $N\!\times\!N$ unitary matrices used for the standard transfer matrix parametrization, we calculate their probability distributions for the circular orthogonal (COE) and unitary (CUE) Dyson ensembles. In this parametrization, we explicitely compare the COE--CUE distributions with those suitable for quasi--$1d$ conductors and insulators. Then, returning to the usual eigenvalue--eigenvector parametrization of $S$, we study the distributions of the scattering phase shifts. For a quasi--$1d$ metallic system, microscopic simulations show that the phase sift density and correlation functions are close to those of the circular ensembles. When quasi--$1d$ longitudinal localization breaks $S$ into two uncorrelated reflection matrices, the phase shift form factor $b(k)$ exhibits a crossover from a behavior characteristic of two uncoupled COE--CUE (small $k$) to a single COE--CUE behavior (large $k$). Outside quasi--one dimension, we find that the phase shift density is no longer uniform and $S$ remains nonzero after disorder averaging. We use perturbation theory to calculate the deviations to the isotropic Dyson distributions. When the electron dynamics is noComment: 39 pages, 14 figures available under request, RevTex, IPNO/TH 94-6

Metal-insulator transition in three dimensional Anderson model: universal scaling of higher Lyapunov exponents

Numerical studies of the Anderson transition are based on the finite-size scaling analysis of the smallest positive Lyapunov exponent. We prove numerically that the same scaling holds also for higher Lyapunov exponents. This scaling supports the hypothesis of the one-parameter scaling of the conductance distribution. From the collected numerical data for quasi one dimensional systems up to the system size 24 x 24 x infinity we found the critical disorder 16.50 < Wc < 16.53 and the critical exponent 1.50 < \nu < 1.54. Finite-size effects and the role of irrelevant scaling parameters are discussed.Comment: 4 pages, 2 figure

Quantum transmission in disordered insulators: random matrix theory and transverse localization

We consider quantum interferences of classically allowed or forbidden electronic trajectories in disordered dielectrics. Without assuming a directed path approximation, we represent a strongly disordered elastic scatterer by its transmission matrix ${\bf t}$. We recall how the eigenvalue distribution of ${\bf t.t}^{\dagger}$ can be obtained from a certain ansatz leading to a Coulomb gas analogy at a temperature $\beta^{-1}$ which depends on the system symmetries. We recall the consequences of this random matrix theory for quasi--$1d$ insulators and we extend our study to microscopic three dimensional models in the presence of transverse localization. For cubes of size $L$, we find two regimes for the spectra of ${\bf t.t}^{\dagger}$ as a function of the localization length $\xi$. For $L / \xi \approx 1 - 5$, the eigenvalue spacing distribution remains close to the Wigner surmise (eigenvalue repulsion). The usual orthogonal--unitary cross--over is observed for {\it large} magnetic field change $\Delta B \approx \Phi_0 /\xi^2$ where $\Phi_0$ denotes the flux quantum. This field reduces the conductance fluctuations and the average log--conductance (increase of $\xi$) and induces on a given sample large magneto--conductance fluctuations of typical magnitude similar to the sample to sample fluctuations (ergodic behaviour). When $\xi$ is of the order of theComment: Saclay-S93/025 Email: [email protected]

Ground state of a partially melted Wigner molecule

We consider three spinless fermions free to move on 2d square lattice with periodic boundary conditions and interacting via a U/r Coulomb repulsion. When the Coulomb energy to kinetic energy ratio r_s is large, a rigid Wigner molecule is formed. As r_s decreases, we show that melting proceeds via an intermediate regime where a floppy two particle molecule coexists with a partially delocalized particle. A simple ansatz is given to describe the ground state of this mesoscopic solid-liquid regime.Comment: to appear in Europhysics Letter

Quantum interference and the spin orbit interaction in mesoscopic normal-superconducting junctions

We calculate the quantum correction to the classical conductance of a disordered mesoscopic normal-superconducting (NS) junction in which the electron spatial and spin degrees of freedom are coupled by an appreciable spin orbit interaction. We use random matrix theory to describe the scattering in the normal part of the junction and consider both quasi-ballistic and diffusive junctions. The dependence of the junction conductance on the Schottky barrier transparency at the NS interface is also considered. We find that the quantum correction is sensitive to the breaking of spin rotation symmetry even when the junction is in a magnetic field and time reversal symmetry is broken. We demonstrate that this sensitivity is due to quantum interference between scattering processes which involve electrons and holes traversing closed loops in the same direction. We explain why such processes are sensitive to the spin orbit interaction but not to a magnetic field. Finally we consider the effect of the spin orbit interaction on the phenomenon of ``reflectionless tunnelling.''Comment: Revised version, one new figure and revised text. This is the final version which will appear in Journal de Physqiue 1. Latex plus six postscript figure

Interacting electron systems between Fermi leads: effective one-body transmissions and correlation clouds

In order to extend the Landauer formulation of quantum transport to correlated fermions, we consider a spinless system in which charge carriers interact, connected to two reservoirs by non-interacting one-dimensional leads. We show that the mapping of the embedded many-body scatterer onto an effective one-body scatterer with interaction-dependent parameters requires to include parts of the attached leads where the interacting region induces power law correlations. Physically, this gives a dependence of the conductance of a mesoscopic scatterer upon the nature of the used leads which is due to electron interactions inside the scatterer. To show this, we consider two identical correlated systems connected by a non-interacting lead of length $L\_\mathrm{C}$. We demonstrate that the effective one-body transmission of the ensemble deviates by an amount $A/L\_\mathrm{C}$ from the behavior obtained assuming an effective one-body description for each element and the combination law of scatterers in series. $A$ is maximum for the interaction strength $U$ around which the Luttinger liquid becomes a Mott insulator in the used model, and vanishes when $U \to 0$ and $U \to \infty$. Analogies with the Kondo problem are pointed out.Comment: 5 pages, 6 figure

From the Fermi glass towards the Mott insulator in one dimension: Delocalization and strongly enhanced persistent currents

When a system of spinless fermions in a disordered mesoscopic ring becomes instable between the inhomogeneous configuration driven by the random potential (Anderson insulator) and the homogeneous one driven by repulsive interactions (Mott insulator), the persistent current can be enhanced by orders of magnitude. This is illustrated by a study of the change of the ground state energy under twisted boundary conditions using the density matrix renormalization group algorithm.Comment: 4 pages, 5 figures; RevTe

Intermediate Regime between the Fermi Glass and the Mott Insulator in one Dimension

We consider the ground state reorganization driven by an increasing nearest neighbor repulsion U for spinless fermions in a strongly disordered ring. When U -> 0, the electrons form a glass with Anderson localized states. At half filling, a regular array of charges (Mott insulator) is pinned by the random substrate when U -> \infty. Between those two insulating limits, we show that there is an intermediate regime where the electron glass becomes more liquid before crystallizing. The liquid-like behavior of the density-density correlation function is accompanied by an enhancement of the persistent current.Comment: 5 pages, Latex, uses moriond.sty (included), Contribution to the Proceedings of the Rencontres de Moriond 199

Long-Range Energy-Level Interaction in Small Metallic Particles

We consider the energy level statistics of non-interacting electrons which diffuse in a $d$-dimensional disordered metallic conductor of characteristic Thouless energy $E_c.$ We assume that the level distribution can be written as the Gibbs distribution of a classical one-dimensional gas of fictitious particles with a pairwise additive interaction potential $f(\varepsilon ).$ We show that the interaction which is consistent with the known correlation function of pairs of energy levels is a logarithmic repulsion for level separations $\varepsilon <E_c,$ in agreement with Random Matrix Theory. When $\varepsilon >E_c,$ $f(\varepsilon )$ vanishes as a power law in $\varepsilon /E_c$ with exponents $-{1 \over 2},-2,$ and $-{3 \over 2}$ for $d=1,2,$ and 3, respectively. While for $d=1,2$ the energy-level interaction is always repulsive, in three dimensions there is long-range level attraction after the short-range logarithmic repulsion.Comment: Saclay-s93/014 Email: [email protected] [2017: missing figure included

Delocalization effects and charge reorganizations induced by repulsive interactions in strongly disordered chains

We study the delocalization effect of a short-range repulsive interaction on the ground state of a finite density of spinless fermions in strongly disordered one dimensional lattices. The density matrix renormalization group method is used to explore the charge density and the sensitivity of the ground state energy with respect to the boundary condition (the persistent current) for a wide range of parameters (carrier density, interaction and disorder). Analytical approaches are developed and allow to understand some mechanisms and limiting conditions. For weak interaction strength, one has a Fermi glass of Anderson localized states, while in the opposite limit of strong interaction, one has a correlated array of charges (Mott insulator). In the two cases, the system is strongly insulating and the ground state energy is essentially invariant under a twist of the boundary conditions. Reducing the interaction strength from large to intermediate values, the quantum melting of the solid array gives rise to a more homogeneous distribution of charges, and the ground state energy changes when the boundary conditions are twisted. In individual chains, this melting occurs by abrupt steps located at sample-dependent values of the interaction where an (avoided) level crossing between the ground state and the first excitation can be observed. Important charge reorganizations take place at the avoided crossings and the persistent currents are strongly enhanced around the corresponding interaction value. These large delocalization effects become smeared and reduced after ensemble averaging. They mainly characterize half filling and strong disorder, but they persist away of this optimal condition.Comment: 18 pages, 15 figures, accepted for publication in Eur. Phys. J.