Anderson Model Of Localization Research Articles

Localization and Directionality of Surface Transport in Bi2Te3 Ordered 3D Nanonetworks.

The resistance of an ordered 3D-Bi2Te3 nanowire nanonetwork was studied at low temperatures. Below 50 K the increase in resistance was found to be compatible with the Anderson model for localization, considering that conduction takes place in individual parallel channels across the whole sample. Angle-dependent magnetoresistance measurements showed a distinctive weak antilocalization characteristic with a double feature that we could associate with transport along two perpendicular directions, dictated by the spatial arrangement of the nanowires. The coherence length obtained from the Hikami-Larkin-Nagaoka model was about 700 nm across transversal nanowires, which corresponded to approximately 10 nanowire junctions. Along the individual nanowires, the coherence length was greatly reduced to about 100 nm. The observed localization effects could be the reason for the enhancement of the Seebeck coefficient observed in the 3D-Bi2Te3 nanowire nanonetwork compared to individual nanowires.

Statistics of Green's functions on a disordered Cayley tree and the validity of forward scattering approximation

The accuracy of the forward scattering approximation for two-point Green's functions of the Anderson localization model on the Cayley tree is studied. A relationship between the moments of the Green's function and the largest eigenvalue of the linearized transfer-matrix equation is proved in the framework of the supersymmetric functional-integral method. The new large-disorder approximation for this eigenvalue is derived and its accuracy is established. Using this approximation the probability distribution of the two-point Green's function is found and compared with that in the forward scattering approximation (FSA). It is shown that FSA overestimates the role of resonances and thus the probability for the Green's function to be significantly larger than its typical value. The error of FSA increases with increasing the distance between points in a two-point Green's function.

Dynamical phases in a ``multifractal'' Rosenzweig-Porter model

We consider the static and the dynamical phases in a Rosenzweig-Porter (RP) random matrix ensemble with a distribution of off-diagonal matrix elements of the form of the large-deviation ansatz. We present a general theory of survival probability in such a random-matrix model and show that the averaged survival probability may decay with time as a simple exponent, as a stretch-exponent and as a power-law or slower. Correspondingly, we identify the exponential, the stretch-exponential and the frozen-dynamics phases. As an example, we consider the mapping of the Anderson localization model on Random Regular Graph onto the RP model and find exact values of the stretch-exponent \kappaκ in the thermodynamic limit. As another example we consider the logarithmically-normal RP random matrix ensemble and find analytically its phase diagram and the exponent \kappaκ. Our theory allows to describe analytically the finite-size multifractality and to compute the critical length with the exponent \nu_{MF}=1νMF=1 associated with it.

Large deviations of the Lyapunov exponent in 2D matrix Langevin dynamics with applications to one-dimensional Anderson localization models

For the 2D matrix Langevin dynamics that correspond to the continuous-time limit of the products of some 2 × 2 random matrices, the finite-time Lyapunov exponent can be written as an additive functional of the associated Riccati process submitted to some Langevin dynamics on the infinite periodic ring. Its large deviations properties can thus be analyzed from two points of view that are equivalent in the end by consistency but give different perspectives. In the first approach, one starts from the large deviations at level 2.5 for the joint probability of the empirical density and empirical current of the Riccati process, and one then performs the appropriate Euler–Lagrange optimization in order to compute the cumulant generating function of the Lyapunov exponent. In the second approach, this cumulant generating function is obtained from the spectral analysis of the appropriate tilted Fokker–Planck operator. The associated conditioned process obtained via the generalization of Doob’s h-transform allows one to clarify the equivalence with the first approach. Finally, applications to one-dimensional Anderson localization models are described in order to explicitly obtain the first cumulants of the finite-size Lyapunov exponent.

Randomness-assisted exponential hierarchies

Inspired by the localization phenomenon in condensed matter systems, we explore constructions in the theory space of multiple scalar fields, in which exponentially suppressed couplings could originate from random parameters. In particular, we find a new class of non-local theory space models, in which scalar fields at non-adjacent sites interact with each other but with strengths decaying exponentially with the site separation. Such a model could have very different localization properties, compared to the local theory space scenarios with only nearest-site interactions, based on the original Anderson localization model. More specifically, we find that a particular non-local interaction pattern leads to bi-localization of the two lightest eigenstates. Exponential localization (and thus exponentially suppressed couplings) then emerges only and immediately when randomness is introduced, no matter how tiny it is. We discuss variants of the model and possible UV completions as well.

Strong disorder RG approach – a short review of recent developments

The Strong Disorder RG approach for random systems has been extended in many new directions since our previous review of 2005 [Phys. Rep. 412, 277]. The aim of the present colloquium paper is thus to give an overview of these various recent developments. In the field of quantum disordered models, recent progress concern Infinite Disorder Fixed Points for short-ranged models in higher dimensions $d>1$, Strong Disorder Fixed Points for long-ranged models, scaling of the entanglement entropy in critical ground-states and after quantum quenches, the RSRG-X procedure to construct the whole set excited stated and the RSRG-t procedure for the unitary dynamics in Many-Body-Localized Phases, the Floquet dynamics of periodically driven chains, the dissipative effects induced by the coupling to external baths, and Anderson Localization models. In the field of classical disordered models, new applications include the contact process for epidemic spreading, the strong disorder renormalization procedure for general master equations, the localization properties of random elastic networks and the synchronization of interacting non-linear dissipative oscillators.

Statistical properties of the Green function in finite size for Anderson localization models with multifractal eigenvectors

For Anderson localization models with multifractal eigenvectors on disordered samples containing N sites, we analyze in a unified framework the consequences for the statistical properties of the Green function. We focus in particular on the imaginary part of the Green function at coinciding points and study the scaling with the size N of the moments of arbitrary indices q when the broadening follows the scaling . For the standard scaling regime , we find in the two limits and that the moments are governed by the anomalous exponents of individual eigenfunctions, without the assumption of strong correlations between the weights of consecutive eigenstates at the same point. For the non-standard scaling regimes , we obtain that the imaginary Green function follows some Fréchet distribution in the typical region, while rare events are important to obtain the scaling of the moments. We describe the application to the case of Gaussian multifractality and to the case of linear multifractality.

Magnetic, optical, and electron transport properties ofn-type CeO2: Small polarons versus Anderson localization

We report magnetic susceptibility, electrical conductivity and optical absorption of ${\mathrm{Ce}}_{1\ensuremath{-}x}{M}_{x}{\mathrm{O}}_{2}$ where $M$ = Nb,Ta and $0\ensuremath{\le}x\ensuremath{\le}0.03$. The dc conductivity follows a simple thermally activated Arrhenius-type behavior in the $T=70$--700 K range with a change in slope at ${T}^{*}\ensuremath{\approx}155$ K. The high-temperature activation energy shows gradual increase from $\ensuremath{\approx}170$ to 220 meV as the dopant concentration increases. The activation energy of the low-temperature conductivity shows a broad minimum of $\ensuremath{\approx}77$ meV at $x\ensuremath{\approx}0.01$. Electron transport and localization mechanisms are analyzed in the framework of the Holstein small polaron, Anderson localization, and Jahn-Teller distortion models. The fit to the small polaron mobility is dramatically improved when, instead of the longitudinal phonons, the transverse optical phonons are considered in the phonon-assisted electron transport. This serves as an indirect evidence of a strong $4{f}^{1}$ orbital interaction with the oxygen ligands, similar to the case of ${\mathrm{PrO}}_{2}$. Based on comparison of the experimental data to the models, it is proposed that the defect-induced random electric fields make the dominant contribution to the electron localization in donor-doped ceria.

Level repulsion exponent β for many-body localization transitions and for Anderson localization transitions via Dyson Brownian motion

The generalization of the Dyson Brownian motion approach of random matrices to Anderson localization (AL) models (Chalker et al 1996 Phys. Rev. Lett. 77 554) and to many-body localization (MBL) Hamiltonians (Serbyn and Moore 2015 arXiv:1508.07293) is revisited to extract the level repulsion exponent β, where in the delocalized phase governed by the Wigner–Dyson statistics, , in the localized phase governed by the Poisson statistics, and at the critical point. The idea is that the Gaussian disorder variables hi are promoted to Gaussian stationary processes hi(t) in order to sample the disorder stationary distribution with some time correlation τ. The statistics of energy levels can then be studied via Langevin and Fokker–Planck equations. For the MBL quantum spin Hamiltonian with random fields hi, we obtain in terms of the Edwards–Anderson matrix for the same eigenstate m = n and for consecutive eigenstates m = n + 1. For the Anderson localization tight-binding Hamiltonian with random on-site energies hi, we find in terms of the density correlation matrix for consecutive eigenstates m = n + 1, while the diagonal element m = n corresponds to the inverse participation ratio of the eigenstate .

Temperature Dependence of Resistivity of MoSi<sub>2</sub>-Si Composite Thin Films

The temperature dependence of resistivity for thin films of a composite of molybdenum disilicide (MoSi2) and silicon (Si) fabricated by radio frequency magnetron sputtering using a target made of a powder mixture of MoSi2 and Si with molar ratio of Si to Mo of 1:X (2.02X2.55) was analyzed. The temperature dependence could be explained by the multiplicative model consisting of a conduction model similar to the Grüneisen-Bloch model, a modified Anderson localization model for 2.02X2.21 and the modified Anderson localization model for 2.39X2.55 over a wide temperature range.

Partitioning schemes and non-integer box sizes for the box-counting algorithm in multifractal analysis

We compare different partitioning schemes for the box-counting algorithm in the multifractal analysis by computing the singularity spectrum and the distribution of the box probabilities. As model system we use the Anderson model of localization in two and three dimensions. We show that a partitioning scheme which includes unrestricted values of the box size and an average over all box origins leads to smaller error bounds than the standard method using only integer ratios of the linear system size and the box size which was found by Rodriguez et al. (Eur. Phys. J. B 67, 77-82 (2009)) to yield the most reliable results.

Statistics of anomalously localized states at the center of band E = 0 in the one-dimensional Anderson localization model

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.

The role of power-law correlated disorder in the Anderson metal-insulator transition

We study the influence of scale-free correlated disorder on the metal-insulator transition in the Anderson model of localization. We use standard transfer matrix calculations and perform finite-size scaling of the largest inverse Lyapunov exponent to obtain the localization length for respective 3D tight-binding systems. The density of states is obtained from the full spectrum of eigenenergies of the Anderson Hamiltonian. We discuss the phase diagram of the metal-insulator transition and the influence of the correlated disorder on the critical exponents.

Exact solution for eigenfunction statistics at the center-of-band anomaly in the Anderson localization model

An exact solution is found for the problem of the center-of-band ($E=0$) anomaly in the one-dimensional (1D) Anderson model of localization. By deriving and solving an equation for the generating function $\Phi(u,\phi)$ we obtained an exact expression in quadratures for statistical moments $I_{q}=\langle |\psi_{E}({\bf r})|^{2q}\rangle$ of normalized wavefunctions $\psi_{E}({\bf r})$ which show violation of one-parameter scaling and emergence of an additional length scale at $E\approx 0$.

FINITE SIZE SCALING ANALYSIS OF THE ANDERSON TRANSITION

This chapter describes the progress made during the past three decades in the finite size scaling analysis of the critical phenomena of the Anderson transition. The scaling theory of localization and the Anderson model of localization are briefly sketched. The finite size scaling method is described. Recent results for the critical exponents of the different symmetry classes are summarised. The importance of corrections to scaling are emphasised. A comparison with experiment is made, and a direction for future work is suggested.

The Random Phase Property and the Lyapunov Spectrum for Disordered Multi-channel Systems

A random phase property establishing a link between quasi-one-dimensional random Schroedinger operators and full random matrix theory is advocated. Briefly summarized it states that the random transfer matrices placed into a normal system of coordinates act on the isotropic frames and lead to a Markov process with a unique invariant measure which is of geometric nature. On the elliptic part of the transfer matrices, this measure is invariant under the full hermitian symplectic group of the universality class under study. While the random phase property can up to now only be proved in special models or in a restricted sense, we provide strong numerical evidence that it holds in the Anderson model of localization. A main outcome of the random phase property is a perturbative calculation of the Lyapunov exponents which shows that the Lyapunov spectrum is equidistant and that the localization lengths for large systems in the unitary, orthogonal and symplectic ensemble differ by a factor 2 each. In an Anderson-Ando model on a tubular geometry with magnetic field and spin-orbit coupling, the normal system of coordinates is calculated and this is used to derive explicit energy dependent formulas for the Lyapunov spectrum.

Statistics of renormalized on-site energies and renormalized hoppings for Anderson localization in two and three dimensions

For Anderson localization models, there exists an exact real-space renormalization procedure at fixed energy which preserves the Green's functions of the remaining sites [H. Aoki, J. Phys. C 13, 3369 (1980)]. Using this procedure for the Anderson tight-binding model in dimensions $d=2,3$, we study numerically the statistical properties of the renormalized on-site energies $ϵ$ and of the renormalized hoppings $V$ as a function of the linear size $L$. We find that the renormalized on-site energies $ϵ$ remain finite in the localized phase in $d=2,3$ and at criticality $(d=3)$, with a finite density at $ϵ=0$ and a power-law decay $1/{ϵ}^{2}$ at large $|ϵ|$. For the renormalized hoppings in the localized phase, we find: $\text{ln}\text{ }{V}_{L}\ensuremath{\simeq}\ensuremath{-}\frac{L}{{\ensuremath{\xi}}_{loc}}+{L}^{\ensuremath{\omega}}u$, where ${\ensuremath{\xi}}_{loc}$ is the localization length and $u$ a random variable of order one. The exponent $\ensuremath{\omega}$ is the droplet exponent characterizing the strong disorder phase of the directed polymer in a random medium of dimension $1+(d\ensuremath{-}1)$, with $\ensuremath{\omega}(d=2)=1/3$ and $\ensuremath{\omega}(d=3)\ensuremath{\simeq}0.24$. At criticality $(d=3)$, the statistics of renormalized hoppings $V$ is multifractal, in direct correspondence with the multifractality of individual eigenstates and of two-point transmissions. In particular, we measure ${\ensuremath{\rho}}_{typ}\ensuremath{\simeq}1$ for the exponent governing the typical decay $\overline{\text{ln}\text{ }{V}_{L}}\ensuremath{\simeq}\ensuremath{-}{\ensuremath{\rho}}_{typ}\text{ }\text{ln}\text{ }L$, in agreement with previous numerical measures of ${\ensuremath{\alpha}}_{typ}=d+{\ensuremath{\rho}}_{typ}\ensuremath{\simeq}4$ for the singularity spectrum $f(\ensuremath{\alpha})$ of individual eigenfunctions. We also present numerical results concerning critical surface properties.

Publisher’s Note: Scaling in Plateau-to-Plateau Transition: A Direct Connection of Quantum Hall Systems with the Anderson Localization Model [Phys. Rev. Lett.102, 216801 (2009)

Publisher’s Note: Scaling in Plateau-to-Plateau Transition: A Direct Connection of Quantum Hall Systems with the Anderson Localization Model [Phys. Rev. Lett.<b>102</b>, 216801 (2009)

Scaling in Plateau-to-Plateau Transition: A Direct Connection of Quantum Hall Systems with the Anderson Localization Model

The quantum Hall-plateau transition was studied at temperatures down to 1 mK in a random alloy disordered high mobility two-dimensional electron gas. A perfect power-law scaling with kappa=0.42 was observed from 1.2 K down to 12 mK. This perfect scaling terminates sharply at a saturation temperature of Ts approximately 10 mK. The saturation is identified as a finite-size effect when the quantum phase coherence length (Lphi proportional, T(-p/2)) reaches the sample size (W) of millimeter scale. From a size dependent study, Ts proportional, W(-1) was observed and p=2 was obtained. The exponent of the localization length, determined directly from the measured kappa and p, is nu=2.38, and the dynamic critical exponent z=1.

Optimisation of multifractal analysis at the 3D Anderson transition using box-size scaling

We study various box-size scaling techniques to obtain the multifractal properties, in terms of the singularity spectrum f(α), of the critical eigenstates at the metal-insulator transition within the 3-D Anderson model of localisation. The typical and ensemble averaged scaling laws of the generalised inverse participation ratios are considered. In pursuit of a numerical optimisation of the box-scaling technique we discuss different box-partitioning schemes including cubic and non-cubic boxes, use of periodic boundary conditions to enlarge the system and single and multiple origins for the partitioning grid are also implemented. We show that the numerically most reliable method is to divide a system of linear size L equally into cubic boxes of size l for which L/l is an integer. This method is the least numerically expensive while having a good reliability.