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.     

Tailoring Anderson localization by disorder correlations in 1D speckle potentials

We study Anderson localization of single particles in continuous, correlated, one-dimensional disordered potentials. We show that tailored correlations can completely change the energy-dependence of the localization length. By considering two suitable models of disorder, we explicitly show that disorder correlations can lead to a nonmonotonic behavior of the localization length versus energy. Numerical calculations performed within the transfer-matrix approach and analytical calculations performed within the phase formalism up to order three show excellent agreement and demonstrate the effect. We finally show how the nonmonotonic behavior of the localization length with energy can be observed using expanding ultracold-atom gases.     

Disorder-Assisted Error Correction in Majorana Chains

It was recently realized that quenched disorder may enhance the reliability of topological qubits by reducing the mobility of anyons at zero temperature. Here we compute storage times with and without disorder for quantum chains with unpaired Majorana fermions ?? the simplest toy model of a quantum memory. Disorder takes the form of a random site-dependent chemical potential. The corresponding one-particle problem is a one-dimensional Anderson model with disorder in the hopping amplitudes. We focus on the zero-temperature storage of a qubit encoded in the ground state of the Majorana chain. Storage and retrieval are modeled by a unitary evolution under the memory Hamiltonian with an unknown weak perturbation followed by an error-correction step. Assuming dynamical localization of the one-particle problem, we show that the storage time grows exponentially with the system size. We give supporting evidence for the required localization property by estimating Lyapunov exponents of the one-particle eigenfunctions. We also simulate the storage process for chains with a few hundred sites. Our numerical results indicate that in the absence of disorder, the storage time grows only as a logarithm of the system size. We provide numerical evidence for the beneficial effect of disorder on storage times and show that suitably chosen pseudorandom potentials can outperform random ones.     

Convergent expansions for asymptotically degenerate inverse correlation lengths of classical lattice spin systems

We obtain convergent expansions for the inverse correlation length associated with various spin-spin correlation functions for some weakly coupled multicomponent classical lattice spin systems. In terms of the lattice quantum field theory associated with the models the expansions provide a convergent perturbation theory for particle masses which are asymptotically degenerate in the limit of zero coupling.Reseach partially supported by CNPq, Brazil.     

Effects of disorder on the correlation energy of rings of hydrogen atoms

The correlation energy of disordered systems has been calculated by means of second-order Rayleigh-Schrödinger perturbation theory in the M?ller-Plesset partitioning. Rings of hydrogen atoms have been chosen as model systems and the degree of disorder has been varied from complete delocalization to complete localization of the one-particle states. The correlation energy was found to have an extremum at an intermediate degree of disorder, corresponding to incomplete localization.     

A colloidal model system with tunable disorder: Solid-fluid transition and discontinuities in the limit of zero disorder

We study a colloidal model system where disorder can be continuously tuned from no disorder --corresponding to a system that can crystallize-- to large disorder where geometrical frustration occurs. The model system consists of colloidal particles with screened electrostatic repulsion. They can only move on single lines which are parallel and equidistant to each other. We introduce disorder by modulating the particle line density. The system exhibits a solid-to-fluid transition which we study by the structure factor and the temporal evolution of the mean-square distance of nearest neighbors on neighboring lines. A determining feature is the occurrence of discontinuities when disorder is tuned to zero. We observe that the peak height of the pair correlation function in the solid phase does not extrapolate to the value of the perfect crystal. Similarly, the mean interaction energy and the screening length at which the solid-fluid transition occurs seem to be discontinuous when the limit of zero disorder is approached.     

Localization-delocalization transition in chains with long-range correlated disorder

We investigated numerically localization properties of electron eigenstates in a chain with long-range correlated diagonal disorder. A tight-binding one-dimensional model with on-site energies exhibiting long-range correlated disorder (LCD) was used with various disorder strength W. LCD was defined so that it gave a power-law spectral density of the form S(k)αk^-p, where p determines the roughness of the potential landscape. Numerical results on the correlation length ξ of eigenstates shows the existence of the localization-delocalization transition at p=2. It is found that the critical values for disorder strength W_c and also the critical exponent ν for localization length change with the values of p.     

Non-locally correlated disorder and delocalization in one dimension (II). Localization length

We study the delocalization transition in a one-dimensional Dirac fermion system with randomly varying mass by using supersymmetric (SUSY) methods. In a previous paper, we calculated density of states and found that (quasi-) extended states near the band center are enhanced by non-local correlation of the random Dirac mass. Numerical studies support this conclusion. In this paper, we calculate the localization length as a function of the correlation length of the disorder. The result shows that the localization length is an increasing function of the correlation of the random mass.     

一维无序二元固体中电子局域性质的研究

从单电子紧束缚模型的哈密顿量出发，格点能量随机取ε_A和ε_B，只计及格点之间的近程跳跃积分，建立了一维无序二元固体模型. 利用负本征值理论及无限阶微扰理论，对系统电子的本征值和本征态进行了数值计算. 结果表明与一定能量本征值对应的电子波函数只分布在系统的一定范围内，显示了其局域性. 借助传输矩阵方法，计算出电子的局域长度，讨论了局域长度随本征能量和无序度的变化关系，并研究了计入不同范围跳跃积分下，局域长度的变化特征. 关键词： 无序 二元固体 电子态 局域长度     

Noninteracting electrons and the metal-insulator transition in two dimensions with correlated impurities

While standard scaling arguments show that a system of noninteracting electrons in two dimensions and in the presence of uncorrelated disorder is insulating, in this paper we discuss the case where interimpurity correlations are included. We find that for pointlike impurities and an infinite interimpurity correlation length, a mobility edge exists in 2D even if the individual impurity potentials are random. In the uncorrelated system we recover the scaling results, while in the intermediate regime for length scales comparable to the correlation length, the system behaves like a metal but with increasing fluctuations, before strong localization eventually takes over for length scales much larger than the correlation length. In the intermediate regime, the relevant length scale is given by the interimpurity correlation length, with important consequences for high mobility systems.     

Hamiltonian map approach to 1D Anderson model

A one-dimensional diagonal tight binding electronic system is analyzed with the Hamiltonian map approach to study analytically the inverse localization length of an infinite sample. Both the uncorrelated and the dichotomic correlated random potential sequences are considered in the evaluations of the inverse localization length. Analytical expressions for the invariant measure or the angle density distribution are the main motivation of this work in order to derive analytical results. The well-known uncorrelated weak disorder result of the inverse localization length is derived with a clear procedure. In addition, an analytical expression for high disorder is obtained near the band edge. It is found that the inverse localization length goes to 1 in this limit. Following the procedure used in the uncorrelated situation, an analytical expression for the inverse localization length is also obtained for the dichotomic correlated sequence in the small disorder situation.     

Delocalization in one-dimensional tight-binding models with fractal disorder

In the present work, we investigated the correlation-induced localization-delocalization transition in the one-dimensional tight-binding model with fractal disorder. We obtained a phase transition diagram from localized to extended states based on the normalized localization length by controlling the correlation and the disorder strength of the potential. In addition, the transition of the diffusive property of wavepacket dynamics is shown around the critical point.     

Effect of the illumination length on the statistical distribution of the field scattered from one-dimensional random rough surfaces: analytical formulae derived from the small perturbation method

We consider a dielectric plane surface with a local cylindrical perturbation illuminated by a monochromatic plane wave. The perturbation is represented by a random function assuming values with a Gaussian probability density with zero mean value. Outside the perturbation zone, the scattered field can be represented by a superposition of a continuous spectrum of outgoing plane waves. The stationary phase method leads to the asymptotic field, the angular dependence of which is given by the scattering amplitudes of the propagating plane waves. The small perturbation method applied to the Rayleigh integral and the boundary conditions gives a first-order approximation of the scattering amplitudes. We show that the real part and the imaginary part of the scattering amplitudes are Gaussian stochastic variables with zero mean values and unequal variances. The values of variances depend on the length of the perturbation zone. In most cases, the probability density function for the amplitude is a Hoyt distribution and the phase is not uniformly distributed between -π and π. The standard Rayleigh and uniform distributions are obtained for special values of the length and in the case of an infinite illumination length.     

Effect of the illumination length on the statistical distribution of the field scattered from one-dimensional random rough surfaces: analytical formulae derived from the small perturbation method

We consider a dielectric plane surface with a local cylindrical perturbation illuminated by a monochromatic plane wave. The perturbation is represented by a random function assuming values with a Gaussian probability density with zero mean value. Outside the perturbation zone, the scattered field can be represented by a superposition of a continuous spectrum of outgoing plane waves. The stationary phase method leads to the asymptotic field, the angular dependence of which is given by the scattering amplitudes of the propagating plane waves. The small perturbation method applied to the Rayleigh integral and the boundary conditions gives a first-order approximation of the scattering amplitudes. We show that the real part and the imaginary part of the scattering amplitudes are Gaussian stochastic variables with zero mean values and unequal variances. The values of variances depend on the length of the perturbation zone. In most cases, the probability density function for the amplitude is a Hoyt distribution and the phase is not uniformly distributed between –π and π. The standard Rayleigh and uniform distributions are obtained for special values of the length and in the case of an infinite illumination length.     

Conductivity and localization of electron states in one dimensional disordered systems: Further numerical results

The localization of the electron states and dc-conductivity of the one dimensional Anderson model are investigated with various numerical procedures. It is found that the eigenstates are always exponentially localized and that in the center of the band the localization length is proportional to the inverse square of the disorder. The dc-conductivity, as obtained by using the Kubo-Greenwood formula, obeys the central limit theorem for any finite imaginary frequency, with a variance, which is inversely proportional to the squareroot of the number of states contributing to the transport. There is no exponential length dependence of the Kubo-Greenwood conductivity within this model. The conductivity tends to zero only in the limit of vanishing imaginary frequency.     

DNA分子链电子结构特性研究

在单电子紧束缚近似下，建立了一维无序二元DNA分子链模型，计算了链长为2×10⁴个碱基对的DNA分子链的电子态密度、局域化特性，并探讨了碱基对的不同组分、格点能量无序度对电子局域态的影响.结果表明：由于DNA分子链中格点能量无序及碱基对的不同组分的存在，其电子波函数呈现出局域化的特性，而局域长度作为衡量电子局域化程度的一个尺度，受碱基对的组分及格点能量无序度的影响. 关键词： DNA分子链 电子结构 电子局域态 局域长度     

Quantum Harmonic Oscillator Systems with Disorder

We study many-body properties of quantum harmonic oscillator lattices with disorder. A sufficient condition for dynamical localization, expressed as a zero-velocity Lieb-Robinson bound, is formulated in terms of the decay of the eigenfunction correlators for an effective one-particle Hamiltonian. We show how state-of-the-art techniques for proving Anderson localization can be used to prove that these properties hold in a number of standard models. We also derive bounds on the static and dynamic correlation functions at both zero and positive temperature in terms of one-particle eigenfunction correlators. In particular, we show that static correlations decay exponentially fast if the corresponding effective one-particle Hamiltonian exhibits localization at low energies, regardless of whether there is a gap in the spectrum above the ground state or not. Our results apply to finite as well as to infinite oscillator systems. The eigenfunction correlators that appear are more general than those previously studied in the literature. In particular, we must allow for functions of the Hamiltonian that have a singularity at the bottom of the spectrum. We prove exponential bounds for such correlators for some of the standard models.     

Influence of weak nonlinearity on the 1D Anderson model with long-range correlated disorder

We investigate theoretically the nature of the states and the localization properties in a one-dimensional Anderson model with long-range correlated disorder and weak nonlinearity. Using the stationary discrete nonlinear Schrödinger equation, we calculate the disorder-averaged logarithm of the transmittance and the localization length in the fixed input case in a numerically exact manner. Unlike in many previous studies, we strictly fix the intensity of the incident wave and calculate the localization length as a function of other parameters. We also calculate the wave functions in a given disorder configuration. In the linear case, flat phased localized states appear near the bottom of the band and staggered localized states appear near the top of the band, while a continuum of extended states appears near the band center. We find that the focusing Kerr-type nonlinearity enhances the Anderson localization of flat phased states and suppresses that of staggered states. We observe that there exists a perfect symmetry relationship for the localization length between focusing and defocusing nonlinearities. Above a critical value of the strength of nonlinearity, delocalization due to the long-range correlations of disorder is destroyed and all states become localized.     

Ergodicity Breaking and Localization of the Nicolai Supersymmetric Fermion Lattice Model

We investigate dynamics of a supersymmetric fermion lattice model introduced by Nicolai (J Phys A 9:1497–1505, 1976). We show that the Nicolai model has infinitely many local constants of motion for its Heisenberg time evolution, and therefore ergodicity (with respect to thermal equilibrium states) breaks. It has infinitely many degenerated classical ground states. This phenomena is considered as localization at zero temperature. From a viewpoint of perturbation theory, we explain why delocalization is suppressed at zero temperature despite its disorder-free translation-invariant quantum interaction.