Anderson localization and the space-time characteristic of continuum states

A proof of Anderson localization is obtained by ruling out any continuous spectrum on the basis of the space-time characteristic of its states.     

The Repulsion Between Localization Centers in the Anderson Model

In this note we show that, a simple combination of deep results in the theory of random Schrödinger operators yields a quantitative estimate of the fact that the localization centers become far apart, as corresponding energies are close together.     

Weak localization and universal conductance fluctuations in multi-layer graphene

We have performed magneto transport measurements on a multi-layer graphene device fabricated by conventional mechanical exfoliation. Suppression of weak localization (WL) as evidenced by the negative magnetoresistance (NMR) centered at zero field, and reproducible universal conductance fluctuations (UCFs) are observed. Interestingly, it is found that the phase coherence lengths calculated by fitting the observed NMR to conventional WL theory are longer than those determined from fitting the amplitudes of the UCFs to theory in the low temperature regime (T ≤ 8 K). In the high temperature regime (T > 8 K), the phase coherence lengths calculated by fitting the observed NMR to conventional WL theory are shorter than those determined from fitting the amplitudes of the UCFs to theory. Our new results therefore indicate a difference in the electron phase-breaking process between the two models of WL and UCFs in graphene. We speculate that the presence of the capping and bottom graphene layers, which leads the enhancement of disorder in-between, improves the localization condition for WL effect during carrier transportation in the low temperature regime. With increasing temperature, the localization condition for WL in multi-layer graphene becomes much weaker due to strong thermal damping. Therefore, the phase coherence lengths calculated by fitting the observed NMR to conventional WL theory are shorter than those determined from fitting the amplitudes of the UCFs to theory at high temperatures.     

Localization for a Matrix-valued Anderson Model

We study localization properties for a class of one-dimensional, matrix-valued, continuous, random Schrödinger operators, acting on $L^2(\mathbb R)\otimes \mathbb C^NWe study localization properties for a class of one-dimensional, matrix-valued, continuous, random Schr?dinger operators, acting on , for arbitrary N ≥ 1. We prove that, under suitable assumptions on the Fürstenberg group of these operators, valid on an interval , they exhibit localization properties on I, both in the spectral and dynamical sense. After looking at the regularity properties of the Lyapunov exponents and of the integrated density of states, we prove a Wegner estimate and apply a multiscale analysis scheme to prove localization for these operators. We also study an example in this class of operators, for which we can prove the required assumptions on the Fürstenberg group. This group being the one generated by the transfer matrices, we can use, to prove these assumptions, an algebraic result on generating dense Lie subgroups in semisimple real connected Lie groups, due to Breuillard and Gelander. The algebraic methods used here allow us to handle with singular distributions of the random parameters.      

Electromagnetic solitons in a system of graphene planes with Anderson impurities

We consider the Maxwell equations for the electromagnetic-field propagation in a system of graphene planes with Anderson impurities. A phenomenological equation is obtained in the form of an analog of the classical 1 + 1-dimensional sine-Gordon equation. Electrons are considered within the quantum formalism taking into account the dispersion-law variations in the presence of an impurity subsystem. The phenomenological equation is analyzed numerically. It was found that the formation of a forbidden band in the graphene spectrum influenced the propagation of ultrashort optical pulses.     

Weak chaos in the disordered nonlinear Schrödinger chain: Destruction of Anderson localization by Arnold diffusion

The subject of this study is the long-time equilibration dynamics of a strongly disordered one-dimensional chain of coupled weakly anharmonic classical oscillators. It is shown that chaos in this system has a very particular spatial structure: it can be viewed as a dilute gas of chaotic spots. Each chaotic spot corresponds to a stochastic pump which drives the Arnold diffusion of the oscillators surrounding it, thus leading to their relaxation and thermalization. The most important mechanism of equilibration at long distances is provided by random migration of the chaotic spots along the chain, which bears analogy with variable-range hopping of electrons in strongly disordered solids. The corresponding macroscopic transport equations are obtained.     

Zero Energy Anomaly in One-Dimensional Anderson Lattice with Exponentially Correlated Weak Diagonal Disorder

We calculated numerically the localization length of one-dimensional Anderson model with correlated diagonal disorder. For zero energy point in the weak disorder limit, we showed that the localization length changes continuously as the correlation of the disorder increases. We found that higher order terms of the correlation must be included into the current perturbation result in order to give the correct localization length, and to connect smoothly the anomaly at zero correlation with the perturbation result for large correlation.     

Anderson Localization Enhanced Spin Selective Transport of Electrons in DNA

Recent experiments revealed the unusual strong spin effects with high spin selective transmission of electrons in double-stranded DNA. We propose a new mechanism that the strong spin effects could be understood in terms of the combination of the ehiral structure, spin-orbit coupling, and especially spin-dependent Anderson localization. The presence of chiral structure and spin-orbit coupling of DNA induce weak Fermi energy splitting between two spin polarization states. The intrinsic Anderson localization in generic DNA molecules may result in remarkable enhancement of the spin selective transport. In particular, these two spin states with energy splitting have different localization lengths. Spin up/down channel may have shorter/longer localization length so that relatively less/more spin up/down electrons may tunnel through the system. In addition, the strong length dependence of spin selectivity observed in experiments can be naturally understood. Anderson localization enhanced spin selectivity effect may provide a deeper understanding of spin-selective processes in molecular spintronics and biological systems.     

Enhanced localization of waves in one-dimensional random media due to nonlinearity: Fixed input case

We study the influence of nonlinearity on wave localization in one-dimensional random media. Using a discrete nonlinear Schrödinger equation with a random on-site energy term, we calculate the localization length in a numerically exact manner. Unlike in many previous works, we fix the intensity of the incident wave and calculate quantities as a function of other parameters. We find that localization is enhanced due to nonlinearity for the focusing and defocusing nonlinearities. For small nonlinearity, the localization length is a decreasing function of nonlinearity. For sufficiently large nonlinearity, however, the localization length is found to approach a saturation value.     

Zero Energy Anomaly in One-Dimensional Anderson Lattice with Exponentially Correlated Weak Diagonal Disorder

We calculated numerically the localization length of one-dimensional Anderson model with correlated diagonal disorder. For zero energy point in the weak disorder limit, we showed that the localization length changes continuously as the correlation of the disorder increases. We found that higher order terms of the correlation must be included into the current perturbation result in order to give the correct localization length, and to connect smoothly the anomaly at zero correlation with the perturbation result for large correlation.     

Suppression of Anderson localization in disordered metamaterials

We study wave propagation in mixed, 1D disordered stacks of alternating right- and left-handed layers and reveal that the introduction of metamaterials substantially suppresses Anderson localization. At long wavelengths, the localization length in mixed stacks is orders of magnitude larger than for normal structures, proportional to the sixth power of the wavelength, in contrast to the usual quadratic wavelength dependence of normal systems. Suppression of localization is also exemplified in long-wavelength resonances which largely disappear when left-handed materials are introduced.     

Remarkable enhancement of terahertz conductivity of graphene tuned by periodic gate voltage

Optical conductivity of a graphene sheet subject to a one-dimensional cosinusoidal potential is studied. We find that the optical conductivity can be tuned by the strength of the superlattice potential. When this strength is within a certain range the terahertz conductivity can be enhanced by two orders of magnitude.     

Commensurability effects in one-dimensional Anderson localization: Anomalies in eigenfunction statistics

The one-dimensional (1d) Anderson model (AM), i.e. a tight-binding chain with random uncorrelated on-site energies, has statistical anomalies at any rational point , where a is the lattice constant and λ_E is the de Broglie wavelength. We develop a regular approach to anomalous statistics of normalized eigenfunctions ψ(r) at such commensurability points. The approach is based on an exact integral transfer-matrix equation for a generating function Φ_r(u, ?) (u and ? have a meaning of the squared amplitude and phase of eigenfunctions, r is the position of the observation point). This generating function can be used to compute local statistics of eigenfunctions of 1d AM at any disorder and to address the problem of higher-order anomalies at with q > 2. The descender of the generating function P_r(?)≡Φ_r(u=0,?) is shown to be the distribution function of phase which determines the Lyapunov exponent and the local density of states.In the leading order in the small disorder we derived a second-order partial differential equation for the r-independent (“zero-mode”) component Φ(u, ?) at the E = 0 () anomaly. This equation is nonseparable in variables u and ?. Yet, we show that due to a hidden symmetry, it is integrable and we construct an exact solution for Φ(u, ?) explicitly in quadratures. Using this solution we computed moments I_m = N〈∣ψ∣^2m〉 (m ? 1) for a chain of the length N → ∞ and found an essential difference between their m-behavior in the center-of-band anomaly and for energies outside this anomaly. Outside the anomaly the “extrinsic” localization length defined from the Lyapunov exponent coincides with that defined from the inverse participation ratio (“intrinsic” localization length). This is not the case at the E = 0 anomaly where the extrinsic localization length is smaller than the intrinsic one. At E = 0 one also observes an anomalous enhancement of large moments compatible with existence of yet another, much smaller characteristic length scale.     

Anomalous localization in low-dimensional systems with correlated disorder

This review presents a unified view on the problem of Anderson localization in one-dimensional weakly disordered systems with short-range and long-range statistical correlations in random potentials. The following models are analyzed: the models with continuous potentials, the tight-binding models of the Anderson type, and various Kronig–Penney models with different types of perturbations. Main attention is paid to the methods of obtaining the localization length in dependence on the controlling parameters of the models. Specific interest is in an emergence of effective mobility edges due to certain long-range correlations in a disorder. The predictions of the theoretical and numerical analysis are compared to recent experiments on microwave transmission through randomly filled waveguides.     

Anderson localization of a Bose-Einstein condensate in a 3D random potential

We study the effect of Anderson localization on the expansion of a Bose-Einstein condensate, released from a harmonic trap, in a 3D random potential. We use scaling arguments and the self-consistent theory of localization to show that the long-time behavior of the condensate density is controlled by a single parameter equal to the ratio of the mobility edge and the chemical potential of the condensate. We find that the two critical exponents of the localization transition determine the evolution of the condensate density in time and space.     

Anderson localization of expanding Bose-Einstein condensates in random potentials

We show that the expansion of an initially confined interacting 1D Bose-Einstein condensate can exhibit Anderson localization in a weak random potential with correlation length sigma(R). For speckle potentials the Fourier transform of the correlation function vanishes for momenta k>2/sigma(R) so that the Lyapunov exponent vanishes in the Born approximation for k>1/sigma(R). Then, for the initial healing length of the condensate xi(in)>sigma(R) the localization is exponential, and for xi(in)相似文献   

大尺度介观电学输运在纳米结构石墨烯中的实现

Anderson局域化是量子波动性导致的最重要的物理现象之一。Anderson局域化理论原是对电子体系提出的,但是由于电子波动性只在很小的范围内(即相位相干长度内)有效,使得Anderson局域化的观测成为一个难点。在文章中,作者报道了在纳米结构石墨烯中首次观测到的二维Anderson强局域化现象。更重要的是作者找到了使电子相位相干长度增长至少一个量级的方法,使得电子的相位可以更容易地被操控。作者用尺寸标度方法得到三组局域化长度分别为1.1,2.0和3.4mm。局域化长度随磁场的变化和理论预测符合得非常好。大尺度介观电学输运,表现为并行于二维变程跳跃电导的另一通道。低温下(T<25 K)观测到费米能级附近存在的库仑准能隙抑制了电子与电子间的非弹性散射,从而使得相位相干长度增长到10mm。     

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在文献[1]中,通过分析实验上观察到的测地声模的小密度-电位比与测地声模谐波定级的自洽性,论证了测地声模是在转向点附近一种“蝌蚪状”的定域结构,但没有给出导致这种结构的物理机制.从随机媒质对波传播产生定域化的角度进行了探讨,采用测地声模由漂移波湍流参量激发的产生模型,计算了随机定域化导致的也是文献[1]所关注的相干长度.讨论了与实验观测比较的有关问题.