Two interacting particles in a random potential

We study the scaling of the localization length of two interacting bosons in a one-dimensional random lattice with the single particle localization length. We consider the short-range interaction assuming that the particles interact when located both on the same site. We discuss several regimes, among them one interesting weak Fock space disorder regime. In this regime we obtain a weak logarithmic scaling law. Numerical benchmark data support the absence of any strong enhancement of the two particle localization length.     

Scaling theory of quantum resistance distributions in disordered systems

We have derived explicitly, the large scale distribution of quantum Ohmic resistance of a disordered one-dimensional conductor. We show that in the thermodynamic limit this distribution is characterized by two independent parameters for strong disorder, leading to a two-parameter scaling theory of localization. Only in the limit of weak disorder we recover single parameter scaling, consistent with existing theoretical treatments.     

Electron delocalization and multifractal scaling in electrified random chains

The electron localization property of a random chain changing under the influence of a constant electric field has been studied. We have adopted the multifractal scaling formalism to explore the possible localization behavior in the system. We observe that the possible localization behavior with the increase of the electric field is not systematic and shows strong instabilities associated with the local probability variation over the length of the chain. The multifractal scaling study captures the localization aspects along with the strong instability when the electric field is changed by infinitesimal steps for a reasonably large system size.     

Anderson transition in two-dimensional systems with spin-orbit coupling

We report a numerical investigation of the Anderson transition in two-dimensional systems with spin-orbit coupling. An accurate estimate of the critical exponent nu for the divergence of the localization length in this universality class has to our knowledge not been reported in the literature. Here we analyze the SU(2) model. We find that for this model corrections to scaling due to irrelevant scaling variables may be neglected permitting an accurate estimate of the exponent nu=2.73+/-0.02.     

Analytical realization of finite-size scaling for Anderson localization: Is there a transition in the 2D case?

Roughly half the numerical investigations of the Anderson transition are based on consideration of an associated quasi-1D system and postulation of one-parameter scaling for the minimal Lyapunov exponent. If this algorithm is taken seriously, it leads to unambiguous prediction of the 2D phase transition. The transition is of the Kosterlitz-Thouless type and occurs between exponential and power law localization (Pichard and Sarma, 1981). This conclusion does not contradict numerical results if raw data are considered. As for interpretation of these data in terms of one-parameter scaling, this is inadmissible: the minimal Lyapunov exponent does not obey any scaling. A scaling relation is valid not for a minimal, but for some effective Lyapunov exponent whose dependence on the parameters is determined by the scaling itself. If finite-sizedd scaling is based on the effective Lyapunov exponent, the existence of the 2D transition becomes indefinite, but still rather probable. Interpretation of the results in terms of the Gell-Mann-Low equation is also given.     

Numerical study of the localization length critical index in a network model of plateau-plateau transitions in the quantum Hall effect

We calculate numerically the localization length critical index within the Chalker-Coddington model of the plateau-plateau transitions in the quantum Hall effect. We report a finite-size scaling analysis using both the traditional power-law corrections to the scaling function and the inverse logarithmic ones, which provided a more stable fit resulting in the localization length critical index ν = 2.616 ± 0.014. We observe an increase of the critical exponent ν with the system size, which is possibly the origin of discrepancies with early results obtained for smaller systems.     

Scaling and localization lengths of a topologically disordered system

We consider a noninteracting disordered system designed to model particle diffusion, relaxation in glasses, and impurity bands of semiconductors. Disorder originates in the random spatial distribution of sites. We find strong numerical evidence that this model displays the same universal behavior as the standard Anderson model. We use finite-size scaling to find the localization length as a function of energy and density, including localized states away from the delocalization transition. Results at many energies all fit onto the same universal scaling curve.     

Comment on the paper “Finite-size scaling from the self-consistent theory of localization” by I. M. Suslov

In the recent paper [1], a new scaling theory of electron localization was proposed. We show that numerical data for the quasi-one-dimensional Anderson model do not support predictions of this theory.     

Sum-rule studies for electrons in small metal particles

The electric dipole-matrix element of electrons in a small metallic grain is considered using sum rules and scaling concepts. Applying level statistical arguments a universal scaling form for the dipole-matrix element is obtained which reproduces the previously calculated frequency dependences in the ballistic, diffusive and localization regimes.     

Statistical properties of the one dimensional Anderson model relevant for the nonlinear Schrödinger equation in a random potential

The statistical properties of overlap sums of groups of four eigenfunctions of the Anderson model for localization as well as combinations of four eigenenergies are computed. Some of the distributions are found to be scaling functions, as expected from the scaling theory for localization. These enable to compute the distributions in regimes that are otherwise beyond the computational resources. These distributions are of great importance for the exploration of the nonlinear Schrödinger equation (NLSE) in a random potential since in some explorations the terms we study are considered as noise and the present work describes its statistical properties.     

Scaling behavior of disordered lattice fermions in two dimensions

We propose a lattice model for Dirac fermions which allows us to break the degeneracy of the node structure. In the presence of a random gap we analyze the scaling behavior of the localization length as a function of the system width within a numerical transfer-matrix approach. Depending on the strength of the randomness, there are different scaling regimes for weak, intermediate and strong disorder. These regimes are separated by transitions that are characterized by one-parameter scaling.     

Amplitude scaling behavior of band center states of Frenkel exciton chains with correlated off-diagonal disorder

We report the amplitude scaling behavior of Frenkel exciton chains with nearest-neighbor correlated off-diagonal random interactions. The band center spectrum and its localization properties are investigated through the integrated density of states and the inverse localization length. The correlated random interactions are produced through a binary sequence similar to the interactions in spin glass chains. We produced sets of data with different interaction strength and “wrong” sign concentrations that collapsed after scaling to the predictions of a theory developed earlier for Dirac fermions with random-varying mass. We found good agreement as the energy approaches the band center for a wide range of concentrations. We have also established the concentration dependence of the lowest order expansion coefficient of the scaling amplitudes for the correlated case. The correlation causes unusual behavior of the spectra, i.e., deviations from the Dyson-type singularity.     

Anisotropic localization behavior of graphene in the presence of diagonal and off-diagonal disorders

Anisotropic localization of Dirac fermions in graphene along both the x and y axes was studied using the transfer-matrix method. The two-parameter scaled behavior around the Dirac points was observed along the x axis with off-diagonal disorder. In contrast, the electronic state along the y axis with armchair edges was delocalized, which can be described well by single parameter scaling theory. This implies that the breakdown of the single-parameter scaling is related to the zigzag edge along the x axis. Furthermore, dimerization induced by the substrate suppresses the two-parameter scaling behavior along the x axis and preserves the delocalized state along the y axis. Our results also demonstrate anisotropic localization in graphene with diagonal disorder that can be tuned by dimerization.     

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.     

Scaling in the disordered quantum antiferromagnetic chain

Numerical calculations on the disordered quantum Heisenberg antiferromagnetic chain yield low-temperature behavior independent of the detailed form of the randomness. A simple scaling interpretation, which makes contact with earlier theoretical work, of these complicated calculations is presented. An analogy with a one-parameter scaling theory of localization is explored.     

Quantum scaling laws in the onset of dynamical delocalization

We study the destruction of dynamical localization experimentally observed in an atomic realization of the kicked rotor by a deterministic Hamiltonian perturbation, with a temporal periodicity incommensurate with the principal driving. We show that the destruction is gradual, with well-defined scaling laws for the various classical and quantum parameters, in sharp contrast to predictions based on the analogy with Anderson localization.     

Anomalous Minimum and Scaling Behavior of Localization Length Near an Isolated Flat Band

Using one‐dimensional tight‐binding lattices and an analytical expression based on the Green's matrix, we show that anomalous minimum of the localization length near an isolated flat band, previously found for evanescent waves in a defect‐free photonic crystal waveguide, is a generic feature and exists in the Anderson regime as well, i.e., in the presence of disorder. Our finding reveals a scaling behavior of the localization length in terms of the disorder strength, as well as a summation rule of the inverse localization length in terms of the density of states in different bands. Most interestingly, the latter indicates the possibility of having two localization minima inside a band gap, if this band gap is formed by two flat bands such as in a double‐sided Lieb lattice.     

Photon localization in amorphous photonic crystal

The photon localization in disordered two-dimensional photonic crystal is studied theoretically. It is found that the mean transmission coefficient in the photonic band decreases exponentially as the disorder degree increases, reflecting the occurrence of Anderson localization. The strength of photon localization can be controlled by tuning the disorder degree in the photonic crystal. We think the variation regular of the transmission coefficient in our disordered system is equivalent to that of the scaling theory of localization. PACS 42.70.Qs; 41.20.Jb; 42.25.Dd     

Kosterlitz-Thouless-Type Met al-Insulator Transit ion in a Double-Plane System with a Random Magnetic Field

We study transport properties in,a coupled double-plane system with one pure and the other random, in the presence of a transverse random magnetic field. The localization.length and conductance of the system are calculated by using the finite-size scaling method combined with transfer matrix technique. We find that in the scaling transformation there is a set of fixed points in a continuous line, indicating that the system undergoes a disorder-driven Kosterlitz-Thouless-type metal-insulator transition.     

Localization transition of the three-dimensional lorentz model and continuum percolation

The localization transition and the critical properties of the Lorentz model in three dimensions are investigated by computer simulations. We give a coherent and quantitative explanation of the dynamics in terms of continuum percolation theory and obtain an excellent matching of the critical density and exponents. Within a dynamic scaling ansatz incorporating two divergent length scales we achieve data collapse for the mean-square displacements and identify the leading corrections to scaling. We provide evidence for a divergent non-Gaussian parameter close to the transition.