Numerical analysis of the Anderson localization

Quantum diffusion induced by small quantum chaos

It is demonstrated that quantum systems classically exhibiting strong and homogeneous chaos in a bounded region of the phase space can induce a global quantum diffusion. As an ideal model system, a small quantum chaos with finite Hilbert space dimension N weakly coupled with M additional degrees of freedom which is approximated by linear systems is proposed. By twinning the system the diffusion process in the additional modes can be numerically investigated without taking the unbounded diffusion space into account explicitly. Even though N is not very large, diffusion occurs in the additional modes as the coupling strength increases if M≥3. If N is large enough, a definite quantum transition to diffusion takes place through a critical subdiffusion characterized by an anomalous diffusion exponent.

Numerical investigation of localization in two-dimensional quasiperiodic mosaic lattice

A one-dimensional lattice model with mosaic quasiperiodic potential is found to exhibit interesting localization properties, e.g. clear mobility edges (Wanget al2020Phys. Rev. Lett.125196604). We generalize this mosaic quasiperiodic model to a two-dimensional version, and numerically investigate its localization properties: the phase diagram from the fractal dimension of the wavefunction, the statistical and scaling properties of the conductance. Compared with disordered systems, our model shares many common features but also exhibits some different characteristics in the same dimensionality and the same universality class. For example, the sharp peak atg∼0of the critical distribution and the largeglimit of the universal scaling functionβresemble those behaviors of three-dimensional disordered systems.

Super-Universality in Anderson Localization

We calculate the effective spatial dimension d_{IR} of electron modes at critical points of 3D Anderson models in various universality classes (O,U,S,AIII). The results are equal within errors, and suggest the super-universal value d_{IR}=2.665(3)≈8/3. The existence of such a unique marker may help identify natural processes driven by Anderson localization, and provide new insight into the spatial geometry of Anderson transitions. The recently introduced d_{IR} is a measure-based dimension of Minkowski-Hausdorff type, designed to characterize probability-induced effective subsets.

Localization and delocalization properties in quasi-periodically-driven one-dimensional disordered systems

Localization and delocalization of quantum diffusion in a time-continuous one-dimensional Anderson model perturbed by the quasiperiodic harmonic oscillations of M colors is investigated systematically, which has been partly reported by a preliminary Letter [H. S. Yamada and K. S. Ikeda, Phys. Rev. E 103, L040202 (2021)2470-004510.1103/PhysRevE.103.L040202]. We investigate in detail the localization-delocalization characteristics of the model with respect to three parameters: the disorder strength W, the perturbation strength ε, and the number of colors, M, which plays the similar role of spatial dimension. In particular, attention is focused on the presence of localization-delocalization transition (LDT) and its critical properties. For M≥3 the LDT exists and a normal diffusion is recovered above a critical strength ε, and the characteristics of diffusion dynamics mimic the diffusion process predicted for the stochastically perturbed Anderson model even though M is not large. These results are compared with the results of discrete-time quantum maps, i.e., the Anderson map and the standard map. Further, the features of delocalized dynamics are discussed in comparison with a limit model which has no static disordered part.

Overactivated transport in the localized phase of the superconductor-insulator transition

Beyond a critical disorder, two-dimensional (2D) superconductors become insulating. In this Superconductor-Insulator Transition (SIT), the nature of the insulator is still controversial. Here, we present an extensive experimental study on insulating NbxSi1-x close to the SIT, as well as corresponding numerical simulations of the electrical conductivity. At low temperatures, we show that electronic transport is activated and dominated by charging energies. The sample thickness variation results in a large spread of activation temperatures, fine-tuned via disorder. We show numerically and experimentally that this originates from the localization length varying exponentially with thickness. At the lowest temperatures, there is an increase in activation energy related to the temperature at which this overactivated regime is observed. This relation, observed in many 2D systems shows that conduction is dominated by single charges that have to overcome the gap when entering superconducting grains.

Presence and absence of delocalization-localization transition in coherently perturbed disordered lattices

A new type of delocalization induced by coherent harmonic perturbations in one-dimensional Anderson-localized disordered systems is investigated. With only a few M frequencies a normal diffusion is realized, but the transition to a localized state always occurs as the perturbation strength is weakened below a critical value. The nature of the transition qualitatively follows the Anderson transition (AT) if the number of degrees of freedom M+1 is regarded as the spatial dimension d. However, the critical dimension is found to be d=M+1=3 and is not d=M+1=2, which should naturally be expected by the one-parameter scaling hypothesis.

Effective Number Theory: Counting the Identities of a Quantum State

Quantum physics frequently involves a need to count the states, subspaces, measurement outcomes, and other elements of quantum dynamics. However, with quantum mechanics assigning probabilities to such objects, it is often desirable to work with the notion of a “total” that takes into account their varied relevance. For example, such an effective count of position states available to a lattice electron could characterize its localization properties. Similarly, the effective total of outcomes in the measurement step of a quantum computation relates to the efficiency of the quantum algorithm. Despite a broad need for effective counting, a well-founded prescription has not been formulated. Instead, the assignments that do not respect the measure-like nature of the concept, such as versions of the participation number or exponentiated entropies, are used in some areas. Here, we develop the additive theory of effective number functions (ENFs), namely functions assigning consistent totals to collections of objects endowed with probability weights. Our analysis reveals the existence of a minimal total, realized by the unique ENF, which leads to effective counting with absolute meaning. Touching upon the nature of the measure, our results may find applications not only in quantum physics, but also in other quantitative sciences.

Critical phenomena of dynamical delocalization in quantum maps: Standard map and Anderson map

Following the paper exploring the Anderson localization of monochromatically perturbed kicked quantum maps [Phys. Rev. E 97, 012210 (2018)2470-004510.1103/PhysRevE.97.012210], the delocalization-localization transition phenomena in polychromatically perturbed quantum maps (QM) is investigated focusing particularly on the dependency of critical phenomena on the number M of the harmonic perturbations, where M+1=d corresponds to the spatial dimension of the ordinary disordered lattice. The standard map and the Anderson map are treated and compared. As the basis of analysis, we apply the self-consistent theory (SCT) of the localization for our systems, taking a plausible hypothesis on the mean-free-path parameter which worked successfully in the analyses of the monochromatically perturbed QMs. We compare in detail the numerical results with the predictions of the SCT by largely increasing M. The numerically obtained index of critical subdiffusion t^{α} (t:time) agrees well with the prediction of one-parameter scaling theory α=2/(M+1), but the numerically obtained critical exponent of localization length significantly deviates from the SCT prediction. Deviation from the SCT prediction is drastic for the critical perturbation strength of the transition: If M is fixed, then the SCT presents plausible prediction for the parameter dependence of the critical value, but its value is 1/(M-1) times smaller than the SCT prediction, which implies existence of a strong cooperativity of the harmonic perturbations with the main mode.

Systematic Quantum Cluster Typical Medium Method for the Study of Localization in Strongly Disordered Electronic Systems

Great progress has been made in recent years towards understanding the properties of disordered electronic systems. In part, this is made possible by recent advances in quantum effective medium methods which enable the study of disorder and electron-electronic interactions on equal footing. They include dynamical mean-field theory and the Coherent Potential Approximation, and their cluster extension, the dynamical cluster approximation. Despite their successes, these methods do not enable the first-principles study of the strongly disordered regime, including the effects of electronic localization. The main focus of this review is the recently developed typical medium dynamical cluster approximation for disordered electronic systems. This method has been constructed to capture disorder-induced localization and is based on a mapping of a lattice onto a quantum cluster embedded in an effective typical medium, which is determined self-consistently. Unlike the average effective medium-based methods mentioned above, typical medium-based methods properly capture the states localized by disorder. The typical medium dynamical cluster approximation not only provides the proper order parameter for Anderson localized states, but it can also incorporate the full complexity of Density-Functional Theory (DFT)-derived potentials into the analysis, including the effect of multiple bands, non-local disorder, and electron-electron interactions. After a brief historical review of other numerical methods for disordered systems, we discuss coarse-graining as a unifying principle for the development of translationally invariant quantum cluster methods. Together, the Coherent Potential Approximation, the Dynamical Mean-Field Theory and the Dynamical Cluster Approximation may be viewed as a single class of approximations with a much-needed small parameter of the inverse cluster size which may be used to control the approximation. We then present an overview of various recent applications of the typical medium dynamical cluster approximation to a variety of models and systems, including single and multiband Anderson model, and models with local and off-diagonal disorder. We then present the application of the method to realistic systems in the framework of the DFT and demonstrate that the resulting method can provide a systematic first-principles method validated by experiment and capable of making experimentally relevant predictions. We also discuss the application of the typical medium dynamical cluster approximation to systems with disorder and electron-electron interactions. Most significantly, we show that in the limits of strong disorder and weak interactions treated perturbatively, that the phenomena of 3D localization, including a mobility edge, remains intact. However, the metal-insulator transition is pushed to larger disorder values by the local interactions. We also study the limits of strong disorder and strong interactions capable of producing moment formation and screening, with a non-perturbative local approximation. Here, we find that the Anderson localization quantum phase transition is accompanied by a quantum-critical fan in the energy-disorder phase diagram.

Anderson Localization of Thermal Phonons Leads to a Thermal Conductivity Maximum

Our elastic model of ErAs disordered GaAs/AlAs superlattices exhibits a local thermal conductivity maximum as a function of length due to exponentially suppressed Anderson-localized phonons. By analyzing the sample-to-sample fluctuations in the dimensionless conductance, g, the transition from diffusive to localized transport is identified as the crossover from the multichannel to single-channel transport regime g ≈ 1. Single parameter scaling is shown to hold in this crossover regime through the universality of the probability distribution of g that is independent of system size and disorder strength.

Critical phenomena of dynamical delocalization in a quantum Anderson map

Using a quantum map version of the one-dimensional Anderson model, the localization-delocalization transition of quantum diffusion induced by coherent dynamical perturbation is investigated in comparison with the quantum standard map. Existence of critical phenomena, which depends on the number of frequency component M, is demonstrated. Diffusion exponents agree with theoretical prediction for the transition, but the critical exponent of the localization length deviates from it with increase in the M. The critical power ε(c) of the normalized perturbation at the transition point remarkably decreases as ε(c)∼(M-1)(-1).

Transversal Anderson localization of sound in acoustic waveguide arrays

We present designs of one-dimensional acoustic waveguide arrays and investigate wave propagation inside. Under the condition of single identical waveguide mode and weak coupling, the acoustic wave motion in waveguide arrays can be modeled with a discrete mode-coupling theory. The coupling constants can be retrieved from simulations or experiments as the function of neighboring waveguide separations. Sound injected into periodic arrays gives rise to the discrete diffraction, exhibiting ballistic or extended transport in transversal direction. But sound injected into randomized waveguide arrays readily leads to Anderson localization transversally. The experimental results show good agreement with simulations and theoretical predictions.

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.

Universal distribution functions in two-dimensional localized systems

We find the conductance distribution function of the two-dimensional Anderson model in the strongly localized limit. The fluctuations of lng grow with lateral size as L1/3 and follow a universal distribution that depends on the type of leads. For narrow leads, it is the Tracy-Widom distribution, which appears in the problem of the largest eigenvalue of random matrices from the Gaussian unitary ensemble and in many other problems like the longest increasing subsequence of a permutation, directed polymers, or polynuclear growth. We also show that for wide leads the conductance follows a related, but different, distribution.