Weak Disorder Limit Research Articles

Electron transport in strongly disordered structures

Using the transfer matrix technique, we investigate the propagation of electron through a two-dimensional disordered sample. We find that the spatial distribution of electrons is homogeneous only in the limit of weak disorder (diffusive transport regime). In the limit of very strong disorder, we identify a narrow channel through which the electron propagates from one side of the sample to the opposite side. Even in this limit, we prove the wave character of the electron propagation.

Coarse graining, fractional moments and the critical slope of random copolymers

For a much-studied model of random copolymer at a selective interface we prove that the slope of the critical curve in the weak-disorder limit is strictly smaller than 1, which is the value given by the annealed inequality. The proof is based on a coarse-graining procedure, combined with upper bounds on the fractional moments of the partition function.

Dissipative oscillations in spatially restricted ecosystems due to long range migration

An ecosystem containing three interacting species is studied using both Mean Field approach and Kinetic Monte Carlo simulations on a lattice substrate. The so called 3rd order LLV model involves birth, death and reaction processes with 3rd order nonlinearities and feedbacks. At the mean field level this system exhibits conservative oscillations; the analytic form of the constant of motion is presented. The stochastic simulations show that the density oscillations disappear for sufficiently large lattices, while they are present locally, on small lattice windows. Introduction of mixing via long range migration in the two reacting species changes this picture. For small migration rates p, the behavior remains as with p = 0 and the system is divided into local asynchronous oscillators. As p increases the system passes through a phase transition and exhibits a weak disorder limit cycle through a supercritical Hopf-like bifurcation. The amplitude of the limit cycle depends on the rate p, on the range of migration r and on the system kinetic rates k1, k2 and k3.

Influence of disorder strength on phase-field models of interfacial growth

We study the influence of disorder strength on the interface roughening process in a phase-field model with locally conserved dynamics. We consider two cases where the mobility coefficient multiplying the locally conserved current is either constant throughout the system (the two-sided model) or becomes zero in the phase into which the interface advances (one-sided model). In the limit of weak disorder, both models are completely equivalent and can reproduce the physical process of a fluid diffusively invading a porous media, where super-rough scaling of the interface fluctuations occurs. On the other hand, increasing disorder causes the scaling properties to change to intrinsic anomalous scaling. In the limit of strong disorder this behavior prevails for the one-sided model, whereas for the two-sided case, nucleation of domains in front of the invading front are observed.

A NOTE ON THE ANALYSIS OF A HETEROGENEOUS ROD-LIKE CHAIN IN DNA MODELING

Two different results concerning the elastic behavior of the heterogeneous worm-like chain (WLC) [D. Bensimon et al., Europhys. Lett.42, 97 (1998)] and rod-like chain (RLC) [P. Nelson, Phys. Rev. Lett.80, 5810 (1998)] are compared. We argue that the RLC is a more suitable model for double-stranded (ds-) DNA. As the hetero-RLC is the basic model for studying sequence-dependent ds-DNA, a rigorous path integral analysis for the effective bending persistence length is performed in the weak disorder limit. The novelty of the paper is in analyzing a path integral on the Lie group SO(3) with random forcing, which supplies a rigorous basis for the analysis of RLC type models.

Effect of weak disorder on the density of states in graphene

The effect of weak potential and bond disorder on the density of states of graphene is studied. By comparing the self-consistent noncrossing approximation on the honeycomb lattice with perturbation theory on the Dirac fermions, we conclude that the linear density of states of pure graphene changes to a nonuniversal power law whose exponent depends on the strength of disorder like $1\ensuremath{-}4g/\sqrt{3}\ensuremath{\pi}{t}^{2}$, with $g$ the variance of the Gaussian disorder and $t$ the hopping integral. This can result in a significant suppression of the exponent of the density of states in the weak-disorder limit. We argue that even a nonlinear density of states can result in a conductivity that is proportional to the number of charge carriers, in accordance with experimental findings.

ELECTRON TRANSPORT THROUGH MULTILEVEL QUANTUM DOT

Quantum transport properties through some multilevel quantum dots sandwiched between two metallic contacts are investigated by the use of Green's function technique. Here, we do parametric calculations, based on the tight-binding model, to study the transport properties through such bridge systems. The electron transport properties are significantly influenced by (a) the number of quantized energy levels in the dots, (b) the dot-to-electrodes coupling strength, (c) the location of the equilibrium Fermi energy E F , and (d) the surface disorder. In the limit of weak-coupling, the conductance (g) shows sharp resonance peaks associated with the quantized energy levels in the dots, while, they get substantial broadening in the strong-coupling limit. The behavior of the electron transfer through these systems becomes much more clearly visible from our study of the current–voltage (I–V) characteristics. In this context, we also describe the noise power of current fluctuations (S) and determine the Fano factor (F) which provides an important information about the electron correlation among the charge carriers. Finally, we explore a novel transport phenomenon by studying the surface disorder effect in which the current amplitude increases with the increase of the surface disorder strength in the strong disorder regime, while, the amplitude decreases in the limit of weak disorder. Such an anomalous behavior is completely opposite to that of bulk disordered system where the current amplitude always decreases with the disorder strength. It is also observed that the current amplitude strongly depends on the system size which reveals the finite quantum size effect.

From the Kubo formula to variable-range hopping

Consider a multichannel closed ring with disorder. In the semiclassical treatment its conductance is given by the Drude formula. Quantum mechanics challenges this result both in the limit of strong disorder (eigenstates are not quantum-ergodic in real space) and in the limit of weak disorder (eigenstates are not quantum ergodic in momentum space). Consequently the analysis of conductance requires going beyond linear response theory, leading to a resistor network picture of transitions between energy levels. We demonstrate that our semilinear response theory provides a firm unified framework from which the ``hopping'' phenomenology of Mott can be derived.

Enhanced carrier scattering rates in dilute magnetic semiconductors with correlated impurities

In III-V dilute magnetic semiconductors such as ${\mathrm{Ga}}_{1\ensuremath{-}x}{\mathrm{Mn}}_{x}\mathrm{As}$, the impurity positions tend to be correlated, which can drastically affect the electronic transport properties of these materials. Within the memory function formalism we have derived a general expression for the current relaxation kernel in spin and charge disordered media and have calculated spin and charge scattering rates in the weak-disorder limit. Using a simple model for impurity clustering, we find a significant enhancement of the charge scattering. The enhancement is sensitive to cluster parameters and may be controllable through postgrowth annealing.

Contact process in heterogeneous and weakly disordered systems

The critical behavior of the contact process (CP) in heterogeneous periodic and weakly disordered environments is investigated using the supercritical series expansion and Monte Carlo (MC) simulations. Phase-separation lines and critical exponents beta (from series expansion) and eta (from MC simulations) are calculated. A general analytical expression for the locus of critical points is suggested for the weak-disorder limit and confirmed by the series expansion analysis and the MC simulations. Our results for the critical exponents show that the CP in heterogeneous environments remains in the directed percolation universality class, while for environments with quenched disorder, the data are compatible with the scenario of continuously changing critical exponents.

Weak disorder in Fibonacci sequences

We study how weak disorder affects the growth of the Fibonacci series. We introduce a family of stochastic sequences that grow by the normal Fibonacci recursion with probability 1-epsilon, but follow a different recursion rule with a small probability epsilon. We focus on the weak disorder limit and obtain the Lyapunov exponent, that characterizes the typical growth of the sequence elements, using perturbation theory. The limiting distribution for the ratio of consecutive sequence elements is obtained as well. A number of variations to the basic Fibonacci recursion including shift, doubling, and copying are considered.

Formation and positioning of nucleosomes: Effect of sequence-dependent long-range correlated structural disorder

The understanding of the long-range correlations (LRC) observed in DNA sequences is still an open and very challenging problem. In this paper, we start reviewing recent results obtained when exploring the scaling properties of eucaryotic, eubacterial and archaeal genomic sequences using the space-scale decomposition provided by the wavelet transform (WT). These results suggest that the existence of LRC up to distances approximately 20-30 kbp is the signature of the nucleosomal structure and dynamics of the chromatin fiber. Actually the LRC are mainly observed in the DNA bending profiles obtained when using some structural coding of the DNA sequences that accounts for the fluctuations of the local double-helix curvature within the nucleosome complex. Because of the approximate planarity of nucleosomal DNA loops, we then study the influence of the LRC structural disorder on the thermodynamical properties of 2D elastic chains submitted locally to mechanical/topological constraint as loops. The equilibrium properties of the one-loop system are derived numerically and analytically in the quite realistic weak-disorder limit. The LRC are shown to favor the spontaneous formation of small loops, the larger the LRC, the smaller the size of the loop. We further investigate the dynamical behavior of such a loop using the mean first passage time (MFPT) formalism. We show that the typical short-time loop dynamics is superdiffusive in the presence of LRC. For displacements larger than the loop size, we use large-deviation theory to derive a LRC-dependent anomalous-diffusion rule that accounts for the lack of disorder self-averaging. Potential biological implications on DNA loops involved in nucleosome positioning and dynamics in eucaryotic chromatin are discussed.

Transport equation for the two-dimensional electron liquid under microwave radiation and a magnetic field: Explanation for the zero-resistance state

A general transport equation for the center of mass motion is constructed to study transports of electronic system under uniform magnetic field and microwave radiation. The equation is applied to study 2D electron system in the limit of weak disorder where negative resistance instability is observed when the radiation field is strong enough. A solution of the transport equation with spontaneous AC current is proposed to explain the experimentally observed Radiation-Induced Zero Resistance State.

Absolutely Continuous Spectra of Quantum Tree Graphs with Weak Disorder

We consider the Laplacian on a rooted metric tree graph with branching number K≥2 and random edge lengths given by independent and identically distributed bounded variables. Our main result is the stability of the absolutely continuous spectrum for weak disorder. A useful tool in the discussion is a function which expresses a directional transmission amplitude to infinity and forms a generalization of the Weyl-Titchmarsh function to trees. The proof of the main result rests on upper bounds on the range of fluctuations of this quantity in the limit of weak disorder.

Nonuniversality of Anderson Localization in Short-Range Correlated Disorder

We provide an analytic theory of Anderson localization on a lattice with a weak short-range correlated disordered potential. Contrary to the general belief, we demonstrate that already next-neighbor statistical correlations in the potential can give rise to strong anomalies in the localization length and the density of states, and to the complete violation of single-parameter scaling. Such anomalies originate in additional symmetries of the lattice model in the limit of weak disorder. The results of numerical simulations are in full agreement with our theory, with no adjustable parameters.

Current flow in random resistor networks: The role of percolation in weak and strong disorder

We study the current flow paths between two edges in a random resistor network on a L X L square lattice. Each resistor has resistance e(ax) , where x is a uniformly distributed random variable and a controls the broadness of the distribution. We find that: (a) The scaled variable u identical with u congruent to L/a(nu) , where nu is the percolation connectedness exponent, fully determines the distribution of the current path length l for all values of u . For u >> 1, the behavior corresponds to the weak disorder limit and l scales as l approximately L, while for u << 1 , the behavior corresponds to the strong disorder limit with l approximately L(d(opt) ), where d(opt) =1.22+/-0.01 is the optimal path exponent. (b) In the weak disorder regime, there is a length scale xi approximately a(nu), below which strong disorder and critical percolation characterize the current path.

Hysteresis and magnetization jumps in theT=0dynamics of spin glasses

We present results from Monte Carlo simulations of hysteresis in the zero-temperature ( T=0 ) dynamics of the Sherrington-Kirkpatrick spin glass model. We study the statistics of magnetization-jumps (denoted as Deltam ) in response to a time-dependent magnetic field H (t) , which increases or decreases with constant increments Delta as H(t)-->H(t)+/-Delta . In particular, we focus on the field dependence of the Deltam -distribution function P(Deltam,H) . We formulate arguments to understand the variation of P ( Deltam,H ) along the hysteresis loop in the weak-disorder limit.

Spin-Hall effect in a disordered two-dimensional electron system

We calculate the spin-Hall conductivity for a two-dimensional electron gas within the self-consistent Born approximation, varying the strength and type of disorder. In the weak disorder limit we find both analytically and numerically a vanishing spin-Hall conductivity even when we allow a momentum dependent scattering. Separating the reactive from the disspative current response, we find the universal value $\sigma^R_{sH} = e/8 \pi$ for the reactive response, which cancels however with the dissipative part $\sigma^D_{sH} = -e/8 \pi$.

Quantum transport properties of two-dimensional systems in disordered magnetic fields with a fixed sign

Quantum transport in disordered magnetic fields is investigated numerically in two-dimensional systems. In particular, the case where the mean and the fluctuation of disordered magnetic fields are of the same order is considered. It is found that in the limit of weak disorder the conductivity exhibits a qualitatively different behavior from that in the conventional random magnetic fields with zero mean. The conductivity is estimated by the equation of motion method and by the two-terminal Landauer formula. It is demonstrated that the conductance stays on the order of $e^2/h$ even in the weak disorder limit. The present behavior can be interpreted in terms of the Drude formula. The Shubnikov-de Haas oscillation is also observed in the weak disorder regime.

Long range disorder and Anderson transition in systems with chiral symmetry

We study the spectral properties of a chiral random banded matrix (chRBM) with elements decaying as a power-law H i j ∼ | i − j | − α . This model is equivalent to a chiral 1D Anderson Hamiltonian with long range power-law hopping. In the weak disorder limit we obtain explicit nonperturbative analytical results for the density of states (DoS) and the two-level correlation function (TLCF) by mapping the chRBM onto a nonlinear σ model. We also put forward, by exploiting the relation between the chRBM at α = 1 and a generalized chiral random matrix model, an exact expression for the above correlation functions. We give compelling analytical and numerical evidence that for this value the chRBM reproduces all the features of an Anderson transition. Finally we discuss possible applications of our results to quantum chromodynamics (QCD).