Inverse localization length for the one-dimensional Anderson model for small disorder

We consider the distribution function P(|ψ|2) of the eigenfunction amplitude at the center-of-band (E = 0) anomaly in the one-dimensional tight-binding chain with weak uncorrelated on-site disorder (the one-dimensional Anderson model). The special emphasis is on the probability of the anomalously localized states (ALS) with |ψ|2 much larger than the inverse typical localization length ℓ0. Using the recently found solution for the generating function Φan(u, ϕ) we obtain the ALS probability distribution P(|ψ|2) at |ψ|2ℓ0 ≫ 1. As an auxiliary preliminary step, we found the asymptotic form of the generating function Φan(u, ϕ) at u ≫ 1 which can be used to compute other statistical properties at the center-of-band anomaly. We show that at moderately large values of |ψ|2ℓ0, the probability of ALS at E = 0 is smaller than at energies away from the anomaly. However, at very large values of |ψ|2ℓ0, the tendency is inverted: it is exponentially easier to create a very strongly localized state at E = 0 than at energies away from the anomaly. We also found the leading term in the behavior of P(|ψ|2) at small |ψ|2 ≪ ℓ−10 and show that it is consistent with the exponential localization corresponding to the Lyapunov exponent found earlier by Kappus and Wegner.

Using the finite-size scaling theory of Fisher and Barber (1972), which is shown to be equivalent to the scaling theory of Abrahams and co-workers (1979), the author investigates the crossover from the linear increase of the residual quantum resistance to the exponential one, as a function of the size of the system. Starting from an explicit tight-binding Hamiltonian with diagonal disorder and using Landauer's formula, he relates the resistance to the trace of a random unimodular matrix product whose eigenvalues are extensively studied. Special care is devoted to length scales smaller than the localisation length, where a classical ohmic behaviour is usually assumed, characterised by a resistivity proportional to the inverse localisation length. He shows numerically that this scaling behaviour is masked by giant fluctuations. He explains, in a simple and pedagogical case, how the actual behaviour of the resistance differs from the scaling prediction by resonance effects. Nevertheless, by summing the fluctuating eigenvalues which are studied in this work for increasing values of the length of the disordered specimen, he gives a more convincing test of the validity of the finite-size concept in one-dimensional Anderson models.

The localization properties of eigenfunctions for two interacting particles in the one-dimensional Anderson model are studied for system sizes up to $N=5000$ sites corresponding to a Hilbert space of dimension $\approx 10^7$ using the Green function Arnoldi method. The eigenfunction structure is illustrated in position, momentum and energy representation, the latter corresponding to an expansion in non-interacting product eigenfunctions. Different types of localization lengths are computed for parameter ranges in system size, disorder and interaction strengths inaccessible until now. We confirm that one-parameter scaling theory can be successfully applied provided that the condition of $N$ being significantly larger than the one-particle localization length $L_1$ is verified. The enhancement effect of the two-particle localization length $L_2$ behaving as $L_2\sim L_1^2$ is clearly confirmed for a certain quite large interval of optimal interactions strengths. Further new results for the interaction dependence in a very large interval, an energy value outside the band center, and different interaction ranges are obtained.

We review the density of states (DOS) and related quantities of quasi one‐dimensional disordered Peierls systems in which fluctuation effects of a backscattering potential play a crucial role. The low‐energy behavior of non‐interacting fermions which are subject to a static random backscattering potential will be described by the strictly one‐dimensional fluctuating gap model (FGM). Recently, the FGM has also been used to explain the pseudogap phenomenon in high‐Tc superconductors. We develop a non‐perturbative method which allows for a simultaneous calculation of the DOS and inverse localization length for an arbitrary given disorder potential by solving a simple initial value problem. In the white noise limit, we recover all known results by solving a Fokker‐Planck equation. For the physically interesting case of finite correlation lengths, we use analytical and numerical methods to show that a complex order parameter leads to a suppression of the DOS, i.e. a pseudogap, and that for a real order parameter this pseudogap is overshadowed by a singularity in the DOS. We will also consider the case of classical phase fluctuations which applies to low temperatures where amplitude fluctuations are frozen out. For this regime we present analytic results for the DOS, the inverse localization length, the specific heat, and the Pauli susceptibility.

We study the nature of electronic states in one-dimensional continuous models with weak correlated disorder. Using a perturbative approach, we compute the inverse localisation length (Lyapunov exponent) up to terms proportional to the fourth power of the potential; this makes possible to analyse the delocalisation transition which takes place when the disorder exhibits specific long-range correlations. We find that the transition consists in a change of the Lyapunov exponent, which switches from a quadratic to a quartic dependence on the strength of the disorder. Within the framework of the fourth-order approximation, we also discuss the different localisation properties which distinguish Gaussian from non-Gaussian random potentials.

Localization of particles in a two-dimensional random potential

Light propagation in a random system of isotropic magnetodielectric layers is studied numerically and analytically. It is shown that if the values of permittivity and permeability are randomly distributed, whereas the characteristic impedance does not change throughout the system, the Lyapunov exponent (the inverse localization length) grows with the angle of incidence and does not depend on the polarization of the incident wave. This independence appears only on the ensemble averaging because in any specific realization the transmission coefficients for s -a ndp-polarized light are different. The numerical results confirm analytical analysis.

The fact that the inverse localization length is normally distributed in disordered onedimensional systems, combined with the delocalizing effects of inelastic scattering, imply that the resistance of thin metallic wires will be a well-behaved random variable in any temperature regime which appears to be accessible by present experimental techniques. In particular, it is shown that the exponentially large fluctuations in the zero-temperature dc resistance found in studies of the Landauer formula can only be of importance in the exponential hopping regime. A condition for these fluctuations to be observed is derived.

In recent years, increasing attention has been paid to the so-called correlated disorder in low-dimensional disordered systems. Interest in this subject is mainly due to two reasons. First, it was found that specific correlations in a disordered potential can result in quite unexpected anomalous properties of scattering. Second, it was shown that such correlations can be relatively easily constructed experimentally, at least in the one-dimensional Anderson model and in Kronig–Penney models of various types. Therefore, it seems to be feasible to fabricate random structures with desired scattering properties, in particular when one needs to suppress or enhance the localization in given frequency windows for scattering electrons or electromagnetic waves. In addition, it was understood that, in many real systems, correlated disorder is an intrinsic property of the underlying structures. One of the most important examples is a DNA chain, for which strong correlations in the potential have been shown to manifest themselves in an anomalous conductance. Thus, the subject of correlated disorder is important both from the theoretical viewpoint, and for various applications in physics.

We analyze the effect of finite memory on the Lyapunov exponent of products of random matrices by considering Markov trials. We study three different cases of physical interest: the one-dimensional Anderson model with correlated random potentials, light propagation in media with correlated random optical indices, and a mimic of the deterministic chaos appearing in dynamical systems with few degrees of freedom. In general the Lyapunov exponent is found to have the same qualitative shape as the inverse of the correlation length of the Markov process. We, however, observe that this rough proportionality fails in some relevant situations. We explain this unexpected behavior in the localization problem by using simple arguments.

We show that at the special energiesE=2cosπp/q, the invariant measure, the Lyapunov exponent, and the density of states can be extended to zero disorder as C∞ functions in the disorder parameter. In particular, we obtain asymptotic series in the disorder parameter. This gives a rigorous proof of the existence of the anomalies originally discovered by Kappus and Wegner and studied by Derrida and Gardner and by Bovier and Klein.

We present a microscopic theory of the transition to and properties of the amorphous solid state of a system of vulcanized (i.e., randomly crosslinked) macromolecules. Our approach invokes a variational hypothesis for the random static density fluctuations characterizing this solid state. The variational parameter is the inverse monomer localization length, which is zero in the liquid state and increases continuously as the mean number of crosslinks exceeds a critical value. The emergent solid is a homogeneous isotropic elastic medium, whose elastic moduli we compute near the transition

Bogomolny's transfer operator method plays a significant role in the study of quantum chaos, along with other well known methods like Gutzwiller's trace formula and the dynamical zeta function, which generalize the Einstein-Brillouin-Keller quantization rule from integrable systems to chaotic systems. According to the theory, the Fredholm determinant of the transfer operator, defined on a Poincaré section of a classical physical system, provides a quantization condition to the energy spectrum of the corresponding quantum system. This study presents two factorization formulas, which relate different quantization conditions defined on different classical trajectory segments. These explicit relations answer the question of why all these classical quantization conditions determine exactly the same energy spectrum of the corresponding quantum systems. As an example, these formulas are illustrated in the equilateral triangular billiard.

The aim of this paper is the construction of numerical tools forthe efficient approximation of transport phenomena in non-autonomous dynamical systems.We focus on transfer operator methods which have been developed in the last yearsfor the treatment of non-autonomous dynamical systems.For instance Froyland et al. [11] proposed a methodfor the approximation of so-called coherent pairs -- these pairs of setsrepresent time-dependent slowly mixing structures -- bythresholding singular vectors of a normalized transfer operator over a fixed time-interval.In principle such transfer operator methods involve long term simulationsof trajectories on the whole state space. In our main result we showthat transport phenomena over a fixed (long) time horizonimply the existence of almost invariant sets over shorter time intervalsif the transport process is slow enough.This fact is used to formulate an algorithm that preselects part of state spaceas a candidate for containing one of the sets of a coherent pair.By this we significantly reduce the related numerical effort.

Using a one-dimensional tight-binding nonlinear Anderson model, we show that the Anderson localization is enhanced by nonlinearity. In the self-focusing case, the localization length is observed to satuate as the nonlinearity increases.