We examine the transport of magnetic energy in a simplified model of a magnetic metamaterial, consisting of a one-dimensional array of split-ring resonators, in the presence of correlated disorder in the resonant frequencies. The computation of the average participation ratio (PR) reveals that on average, the modes for the correlated disorder system are less localized than in the uncorrelated case. The numerical computation of the mean square displacement of an initially localized magnetic excitation for the correlated case shows a substantial departure from the uncorrelated (Anderson-like) case. A long-time asymptotic fit $\langle n^2\rangle \sim t^α$ reveals that, for the uncorrelated system $α\sim 0$, while for the correlated case $α>0$, spanning a whole range of behavior ranging from localization to super-diffusive behavior. The transmission coefficient of a plane wave across a single magnetic dimer reveals the existence of well-defined regions in disorder strength-magnetic coupling space, where unit transmission for some wavevector(s) is possible. This implies, according to the random dimer model (RDM) of Dunlap et al., a degree of mobility. A comparison between the mobilities of the correlated SRR system and the RDM shows that the RDM model has better mobility at low disorder while our correlated SRR model displays better mobility at medium and large disorder.
In a unified approach, we study the transport properties of periodic-on-average bi-layered photonic crystals, metamaterials and electron superlattices. Our consideration is based on the analytical expression for the localization length derived for the case of weakly fluctuating widths of layers, that also takes into account possible correlations in disorder. We analyze how the correlations lead to anomalous properties of transport. In particular, we show that for quarter stack layered media specific correlations can result in a $ω^2$-dependence of the Lyapunov exponent in all spectral bands.
We analyze the effect of microscopic disorder on the macroscopic properties of composite metamaterials and study how weak statistically independent fluctuations of the parameters of structure elements can modify their collective magnetic response and left-handed properties. We demonstrate that even a weak microscopic disorder may lead to a substantial modification of the metamaterial magnetic properties, and 10% deviation in the parameters of the microscopic resonant elements may lead to a substantial suppression of wave propagation in a wide frequency range. A noticeable suppression occurs al…
When the feature size of photonic structures becomes comparable or even smaller than the wavelength of light, the fabrication imperfections inevitably introduce disorder that may eliminate many functionalities of subwavelength photonic devices. Here we suggest a novel concept to achieve a robust bandgap which can endure disorder beyond 30 as a result of the transition from photonic crystals to Mie-resonant metamaterials. By utilizing Mie-resonant metamaterials with high refractive index, we demonstrate photonic waveguides and cavities with strong robustness to position disorder, thus providing…
The localization length has been derived for one-dimensional bi-layered structures with random perturbations in the refractive indices for each type of layers. Main attention is paid to the comparison between conventional materials and those consisting of mixed right-hand and left-hand materials. It is shown that the localization length is described by the universal expression for both cases. The analytical results are confirmed by numerical simulations.
The localization length has been derived for one-dimensional bi-layered structures with random perturbations in the refractive indices for each type of layers. Main attention is paid to the comparison between conventional materials and those consisting of mixed right-hand and left-hand materials. It is shown that the localization length is described by the universal expression for both cases. The analytical results are confirmed by numerical simulations.
We study the problem of non-conventional Anderson localization emerging in bilayer periodic-on-average structures with alternating layers of materials with positive and negative refraction indices $n_a$ and $n_b$. Main attention is paid to the model of the so-called quarter stack with perfectly matched layers (the same unperturbed by disorder impedances, $Z_a=Z_b$, and optical path lengths, $n_ad_a=|n_b| d_b$, with $d_a$, $d_b$ being the thicknesses of basic layers). As was recently numerically discovered, in such structures with weak fluctuations of refractive indices (compositional disorder)…
We explore experimentally a quantum metamaterial based on a superconducting chip with 25 frequency-tunable transmon qubits coupled to a common coplanar resonator. The collective bright and dark modes are probed via the microwave response, i.e., by measuring the transmission amplitude of an external microwave signal. All qubits have individual control and readout lines. Their frequency tunability allows to change the number N of resonantly coupled qubits and also to introduce a disorder in their excitation frequencies with preassigned distributions. While increasing N, we demonstrate the expect…
We study two different versions of the site-diluted Ising model in three dimensions with long-range spatially correlated disorder by Monte Carlo means. We use finite-size scaling techniques to compute the critical exponents of these systems, taking into account the strong scaling-corrections. We find a $ν$ value that is compatible with the analytical predictions.
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.
Problems involving disordered systems are usually analyzed for systems with random disorder. However, there are many systems in which the main disorder involves clusters with correlated differences between their properties and those of the average system. A new approximation, the average trace approximation, is proposed for calculating the diagonal elements of the Green function, and hence the density of states, in such systems. As an example, application of the method to a simple cubic array of harmonic oscillators shows that correlation in the disorder leads to a peak in the low frequency de…
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.
We study the band-centre anomaly in the one-dimensional Anderson model with weak correlated disorder. Our analysis is based on the Hamiltonian map approach; the correspondence between the discrete model and its continuous counterpart is discussed in detail. We obtain analytical expressions of the localisation length and of the invariant measure of the phase variable, valid for energies in a neighbourhood of the band centre. By applying these general results to specific forms of correlated disorder, we show how correlations can enhance or suppress the anomaly at the band centre.
The localization lengths of long-range correlated disordered chains are studied for electronic wavefunctions in the Anderson model and for vibrational states. A scaling theory close to the band edge is developed in the Anderson model and supported by numerical simulations. This scaling theory is mapped onto the vibrational case at small frequencies. It is shown that for small frequencies, unexpectateley the localization length is smaller for correlated than for uncorrelated chains.
We analyse the thermal properties of a harmonic chain with weak correlated disorder. With the use of a perturbative approach we derive analytical expressions for the time-evolution of the chain temperature and of the heat flow when both ends of the chain are coupled to heat baths. Our analytical and numerical results demonstrate that specific long-range correlations of the isotopic disorder can suppress or enhance the vibrational modes in pre-defined frequency windows. In this way one can arrange a frequency-selective heat flow through disordered chains.
The scale invariant properties of wave functions in finite samples of one dimensional random systems with correlated disorder are analyzed. The random dimer model and its generalizations are considered and the wave functions are compared. Generalized entropic localization lengths are introduced in order to characterize the states and compared with their behavior for exponential localization. An acceptable agreement is obtained, however, the exponential form seems to be an oversimplification in the presence of correlated disorder. According to our analysis in the case of the random dimer model …
Localization property in the disordered few-chain DNA systems with a long-range correlation is numerically investigated. We apply the chain system with the correlated disorder in the interchain and/or intrachain hoppings to the simple model of a double strand of DNA. Numerical results for the density of states and the Lyapunov exponent of the wave function in the two- or three-chain models are given. It is found that the correlation effect enhances the localization length (the inverse least nonnegative Lyapunov exponent) around the band center.
The equation for the wave-function localization length in terms of the two-point correlation function of a weak random potential in 1D (F. M. Izrailev and A. A. Krokhin, Phys. Rev. Lett. vol. 82, 4062 (1999)) is rederived using the standard weak-localization theory. I propose a modified algorithm for generation of arbitrary correlated random potentials, and show that numerical calculation of the localization length in a specific correlated disorder in 1D is then in accord with the analytic expression, removing the discrepancy noted in the above Letter.
Tight-binding 1D random system with long-range correlations is studied numerically using the localisation criterium, which represents the number of sites, covered by the wave function. At low degrees of disorder the signs of a mobility edge, predicted in \cite{Izr}, were found. The possibility of exact mobility edge in the system under consideration is discussed.
We study the delocalisation transition which takes places in one-dimensional disordered systems when the random potential exhibits specific long-range correlations. We consider the case of weak disorder; using a systematic perturbative approach, we show how the delocalisation transition brings about a change of the scaling law of the inverse localisation length which ceases to be a quadratic function of the disorder strength and assumes a quartic form when the threshold separating the localised phase from the extended one is crossed.
The tight-binding model with correlated disorder introduced by Izrailev and Krokhin [PRL 82, 4062 (1999)] has been extended to the Kronig-Penney model. The results of the calculations have been compared with microwave transmission spectra through a single-mode waveguide with inserted correlated scatterers. All predicted bands and mobility edges have been found in the experiment, thus demonstrating that any wanted combination of transparent and non-transparent frequency intervals can be realized experimentally by introducing appropriate correlations between scatterers.
We introduce a new approach to analyse the global structure of electronic states in quasi-1D models in terms of the dynamics of a system of parametric oscillators with time-dependent stochastic couplings. We thus extend to quasi-1D models the method previously applied to 1D disordered models. Using this approach, we show that a ``delocalisation transition'' can occur in quasi-1D models with weak disorder with long-range correlations.
Statistical and scaling properties of the Lyapunov exponent for a tight-binding model with the diagonal disorder described by a dichotomic process are considered near the band edge. The effect of correlations on scaling properties is discussed. It is shown that correlations lead to an additional parameter governing the validity of single parameter scaling.
We perform an analytical study of the correspondence between a classical oscillator with frequency perturbed by a coloured noise and the one-dimensional Anderson-type model with correlated diagonal disorder. It is rigorously shown that localisation of electronic states in the quantum model corresponds to exponential divergence of nearby trajectories of the classical random oscillator. We discuss the relation between the localisation length for the quantum model and the rate of energy growth for the stochastic oscillator. Finally, we examine the problem of electron transmission through a finite…
We investigate a tight-binding electronic chain featuring diagonal and off-diagonal disorder, these being modelled through the long-range-correlated fractional Brownian motion. Particularly, by employing exact diagonalization methods, we evaluate how the eigenstate spectrum of the system and its related single-particle dynamics respond to both competing sources of disorder. Moreover, we report the possibility of carrying out efficient end-to-end quantum-state transfer protocols even in the presence of such generalized disorder due to the appearance of extended states around the middle of the b…
We study the critical behavior of the 2D $N$-color Ashkin-Teller model in the presence of random bond disorder whose correlations decays with the distance $r$ as a power-law $r^{-a}$. We consider the case when the spins of different colors sitting at the same site are coupled by the same bond and map this problem onto the 2D system of $N/2$ flavors of interacting Dirac fermions in the presence of correlated disorder. Using renormalization group we show that for $N=2$, a "weakly universal" scaling behavior at the continuous transition becomes universal with new critical exponents. For $N>2$, th…
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 fo…
Distribution of the transmission coefficient T of a long system with a correlated Gaussian disorder is studied analytically and numerically in terms of the generalized Lyapunov exponent (LE) and the cumulants of lnT. The effect of the disorder correlations on these quantities is considered in weak, moderate and strong disorder for different models of correlation. Scaling relations between the cumulants of lnT are obtained. The cumulants are treated analytically within the semiclassical approximation in strong disorder, and numerically for an arbitrary strength of the disorder. A small correlat…
We analyze one-dimensional motion of an overdamped classical particle in the presence of external disorder potential and an arbitrary driving force F. In thermodynamical limit the effective force-dependent mobility mu(F) is self-averaging, although the required system size may be exponentially large for strong disorder. We calculate the mobility mu(F) exactly, generalizing the known results in linear response (weak driving force) and the perturbation theory in powers of the disorder amplitude. For a strong disorder potential with power-law correlations we identify a non-linear regime with a pr…
The localization behavior of the one-dimensional Anderson model with correlated and uncorrelated purely off-diagonal disorder is studied. Using the transfer matrix method, we derive an analytical expression for the localization length at the band center in terms of the pair correlation function. It is proved that for long-range correlated hopping disorder, a localization-delocalization transition occurs at the critical Hurst exponent H_c= 1/2 when the variance of the logarithm of hopping "σ_{\ln(t)}" is kept fixed with the system size N. Based on numerical calculations, finite size scaling rel…
We study the metal-insulator transition in a tight-binding one-dimensional (1D) model with long-range correlated disorder. In the case of diagonal disorder with site energy within $[-\frac{W}{2},\frac{W}{2}]$ and having a power-law spectral density $S(k)\propto k^{-α}$, we investigate the competition between the disorder and correlation. Using the transfer-matrix method and finite-size scaling analysis, we find out that there is a finite range of extended eigenstates for $α>2$, and the mobility edges are at $\pm E_{c}=\pm|2-W/2|$. Furthermore, we find the critical exponent $ν$ of localization …
We study the structure of the electronic states and the transport properties of a Kronig-Penney model with weak compositional and structural disorder. Using a perturbative approach we obtain an analytical expression for the localisation length which is valid for disorder with arbitrary correlations. We show how to generate disorder with self- and cross-correlations and we analyse both the known delocalisation effects of the long-range self-correlations and new effects produced by cross-correlations. We finally discuss how both kinds of correlations alter the transport properties in Kronig-Penn…
We summarize the results of our comprehensive analytical and numerical studies of the effects of polarization on the Anderson localization of classical waves in one-dimensional random stacks. We consider homogeneous stacks composed entirely of normal materials or metamaterials, and also mixed stacks composed of alternating layers of a normal material and metamaterial. We extend the theoretical study developed earlier for the case of normal incidence [A. A. Asatryan et al, Phys. Rev. B 81, 075124 (2010)] to the case of off-axis incidence. For the general case where both the refractive indices a…
Strongly correlated fermions in a crystal or in an optical lattice in the presence of binary alloy disorder are investigated. We employ the statistical dynamical mean-field theory, which incorporates both, local fluctuations due to disorder and local correlations due to interaction, to solve the Anderson-Hubbard model. Localization due to disorder is studied by means of the probability distribution function of the local density of states. We obtain a complete paramagnetic ground state phase diagram consisting of disordered correlated metal, Anderson-Mott insulator, and band insulator.
We study elastic systems such as interfaces or lattices pinned by correlated quenched disorder considering two different types of correlations: generalized columnar disorder and quenched defects correlated as ~ x^{-a} for large separation x. Using functional renormalization group methods, we obtain the critical exponents to two-loop order and calculate the response to a transverse field h. The correlated disorder violates the statistical tilt symmetry resulting in nonlinear response to a tilt. Elastic systems with columnar disorder exhibit a transverse Meissner effect: disorder generates the c…
We investigate localization properties of electron eigenstates in one-dimensional (1d) systems with long-range correlated diagonal disorder. Numerical studies on the localization length $ξ$ of eigenstates demonstrate the existence of the localization-delocalization transition in 1d systems and elucidate non-trivial behavior of $ξ$ as a function of the disorder strength. The critical exponent $ν$ for localization length is extracted for various values of parameters characterizing the disorder, revealing that every $ν$ disobeys the Harris criterion $ν> 2/d$.
We evaluate the mean escape time of overdamped particles over potential barriers in short-correlated quenched Gaussian disorder potentials in one dimension at low temperature. The thermally activated escape is very sensitive to the form of the \emph{tail} of the potential barrier probability distribution. We evaluate this tail by using the optimal fluctuation method. For monotone decreasing autocovariances we reproduce the tail obtained by Lopatin and Vinokur (2001). However, for nonmonotonic autocovariances of the disorder potential we show that the tail changes. It is much higher when the di…
We investigate the explicit renormalization group for fermionic field theoretic representation of two-dimensional random bond Ising model with long-range correlated disorder. We show that a new fixed point appears by introducing a long-range correlated disorder. Such as the one has been observed in previous works for the bosonic ($ϕ^4$) description. We have calculated the correlation length exponent and the anomalous scaling dimension of fermionic fields at this fixed point. Our results are in agreement with the extended Harris criterion derived by Weinrib and Halperin.
A Weyl semimetal denotes an electronic phase of solids in which two bands cross linearly. In this paper we study the effect of a spatially correlated disorder on such a phase. Using a renormalization group analysis, we show that in three dimensions, three scenarios are possible depending on the disorder correlations. A standard transition is recovered for short range correlations. For disorder decaying slower than $1/r^{2}$, the Weyl semimetal is unstable to any weak disorder and no transition persists. In between, a new phase transition occurs. This transition still separates a disordered met…
We investigate the outbreak of anomalous quantum wavefunction amplitudes in a one-dimensional tight-binding lattice featuring correlated diagonal disorder. Such rogue-wave-like behavior is fostered by a competition between localization and mobility. The effective correlation length of the disorder is ultimately responsible for bringing the local disorder strength to a minimum, fuelling the occurrence of extreme events of much higher amplitudes, specially when compared to the case of uncorrelated disorder. Our findings are valid for a class of discrete one-dimensional systems and reveal profoun…
Local ultrafast optical excitation of electron-hole pairs in disordered semiconductors provides the possibility to observe experimentally interaction-assisted propagation of correlated quantum particles in a disordered environment. In addition to the interaction driven delocalization known for the conventional single-band TIP-(two-interacting-particles)-problem the semiconductor model has a richer variety of physical parameters that give rise to new features in the temporal dynamics. These include different masses, correlated vs. anticorrelated disorder for the two particles, and dependence on…
We analyze the impact of a porous medium (structural disorder) on the scaling of the partition function of a star polymer immersed in a good solvent. We show that corresponding scaling exponents change if the disorder is long-range-correlated and calculate the exponents in the new universality class. A notable finding is that star and chain polymers react in qualitatively different manner on the presence of disorder: the corresponding scaling exponents increase for chains and decrease for stars. We discuss the physical consequences of this difference.
Advances in material growth methods have renewed the interest in localization of one-dimensional systems in the presence of scale-free long-range correlated disorder potentials. We analyze the validity of single parameter scaling for the β-function away from the band center, in the presence of correlations. A renormalized disorder strength emerges reducing the regime of validity of the single parameter scaling hypothesis. Analysis of localization lengths and participation ratios leads to correlation dependent critical and fractal exponents, consistent with the extended Harris criterion.
We study the effects of short-range correlated disorder arising from chemical dopants or local lattice distortions, on the ferromagnetism of 3d double exchange systems. For this, we integrate out the carriers and treat the resulting disordered spin Hamiltonian within local random phase approximation, whose reliability is shown by direct comparison with Monte Carlo simulations. We find large scale inhomogeneities in the charge, couplings and spin densities. Compared with the homogeneous case, we obtain larger Curie temperatures ($T_{C}$) and very small spin stiffnesses ($D$). As a result, large…
We observe the emergence of a disorder-induced insulating state in a strongly interacting atomic Fermi gas trapped in an optical lattice. This closed quantum system free of a thermal reservoir realizes the disordered Fermi-Hubbard model, which is a minimal model for strongly correlated electronic solids. In measurements of disorder-induced localization obtained via mass transport, we detect interaction-driven delocalization and localization that persists as the temperature of the gas is raised. These behaviors are consistent with many-body localization, which is a novel paradigm for understand…
We study a canonical many-body-localized (MBL) system with power-law-correlated disorders: $s=\frac{1}{2}$ spin chain in a random magnetic field. The power-law-correlated disorder can control the critical regime between the MBL and thermal (ergodic) phases by varying its exponent, and it let us investigate the MBL transitions in detail. Static-eigenstate and dynamic properties of MBL are studied by numerical methods for systems with various long-range correlations. By using energy-resolved distribution of the participation ratio (PR) and calculating some physical quantities related to localiza…
We investigate the integer quantum Hall system in a two dimensional lattice model with spatially correlated disorder by using the efficient method to calculate the Chern number proposed by Fukui \textit{et al}. Distribution of charge density indicates that the extended states at the center of each Landau band have percolating current paths, which are topologically equivalent to the edge states that exist in a system with boundaries. As increasing the strength of disorder, floating feature is observed in an averaged Hall conductance as a function of filling factor. Its relation to the observed …
We provide an analytic theory of Anderson localization on a lattice with a weak short-range correlated disordered potential. Contrary to the general belief we demonstrate that even next-neighbor statistical correlations in the potential can give rise to strong anomalies in the localization length and the density of states, and to the complete violation of single parameter scaling. Such anomalies originate in additional symmetries of the lattice model in the limit of weak disorder. The results of numerical simulations are in full agreement with our theory, with no adjustable parameters.
We consider the $\pm J$ Ising model on a cubic lattice with a gauge-invariant disorder distribution. Disorder depends on a parameter $β_G$ that plays the role of a chemical potential for the amount of frustration. We study the model at a specific value of the disorder parameter $β_G$, where frustration shows long-range correlations. We characterize the universality class, obtaining accurate estimates of the critical exponents: $ν= 0.655(15)$ and $η_q = 1.05(5)$, where $η_q$ is the overlap susceptibility exponent.
We show that in the one-dimensional (1D) Anderson model long-range correlations within the sequence of on-site potentials may lead to a region of extended states in the vicinity of the band centre, i.e., to a correlation-induced insulator-metal transition. Thus, although still disordered, the 1D system can behave as a conductor.
The magnetic ground state phase diagram of the disordered Hubbard model at half-filling is computed in dynamical mean-field theory supplemented with the spin resolved, typical local density of states. The competition between many-body correlations and disorder is found to stabilize paramagnetic and antiferromagnetic metallic phases at weak interactions. Strong disorder leads to Anderson localization of the electrons and suppresses the antiferromagnetic long-range order. Slater and Heisenberg antiferromagnets respond characteristically different to disorder. The results can be tested with cold …
We consider the Ising model on the square lattice with biaxially correlated random ferromagnetic couplings, the critical point of which is fixed by self-duality. The disorder represents a relevant perturbation according to the extended Harris criterion. Critical properties of the system are studied by large scale Monte Carlo simulations. The correlation length critical exponent, ν=2.005(5), corresponds to that expected in a system with isotropic correlated long-range disorder, whereas the scaling dimension of the magnetization density, x_m=0.1294(7), is somewhat larger than in the pure system.…
Anderson localization has been a subject of intense studies for many years. In this context, we study numerically the influence of long-range correlated disorder on the localization behavior in one dimensional systems. We investigate the localization length and the density of states and compare our numerical results with analytical predictions. Specifically, we find two distinct characteristic behaviors in the vicinity of the band center and at the unperturbed band edge, respectively. Furthermore we address the effect of the intrinsic short-range correlations.
The interplay of strong interaction and strong disorder, as contained in the Anderson-Hubbard model, is addressed using two non-perturbative numerical methods: the Lanczos algorithm in the grand canonical ensemble at zero temperature and Quantum Monte Carlo. We find distinctive evidence for a zero-energy anomaly which is robust upon variation of doping, disorder and interaction strength. Its similarities to, and differences from, pseudogap formation in other contexts, including perturbative treatments of interactions and disorder, classical theories of localized charges, and in the clean Hubba…
Systematic deviations from standard Fermi-liquid behavior have been widely observed and documented in several classes of strongly correlated metals. For many of these systems, mounting evidence is emerging that the anomalous behavior is most likely triggered by the interplay of quenched disorder and strong electronic correlations. In this review, we present a broad overview of such disorder-driven non-Fermi-liquid behavior, and discuss various examples where the anomalies have been studied in detail. We describe both their phenomenological aspects as observed in experiment, and the current the…
We consider 2D Dirac fermions in the presence of three types of disorder: random scalar potential, random gauge potential and random mass with long-range correlations decaying as a power law. Using various methods such as the self-consistent Born approximation (SCBA), renormalization group (RG), the matrix Green function formalism and bosonisation we calculate the density of states and study the full counting statistics of fermionic transport at lower energy. The SCBA and RG show that the random correlated scalar potentials generate an algebraically small energy scale below which the density o…
We demonstrate how supercell implementations of conventional lattice dynamical calculations can be used to determine the extent and nature of disorder-induced broadening in the phonon dispersion spectrum of disordered crystalline materials. The approach taken relies on band unfolding, and is first benchmarked against virtual crystal approximation phonon calculations. The different effects of mass and interaction disorder on the phonon broadening are then presented, focussing on the example of a simple cubic binary alloy. For the mass disorder example, the effect of introducing correlated disor…
