Anomalously localized states in the Anderson model

It is demonstrated that anomalously localized states (ALS) in the Anderson model, being lattice specific, are not related to any of the continuous theories. We identify the spatial structure of ALS on a lattice and calculate their likelihood. Analytical results explain peculiarities in previous numerical simulations.     

Anomalously localised states at the Anderson transition

Non-multifractal critical wave functions at the Anderson transition are numerically investigated for the SU(2) model belonging to the two-dimensional symplectic class. These states can be regarded as anomalously localised states (ALS) at criticality. Giving a quantitative definition of ALS, it has been revealed that the probability to find ALS increases with the system size and remains at a finite value even in the thermodynamic limit. The most probable, namely typical, critical states have the multifractal nature, while its probability measure is zero. In order to understand how ALS affect critical properties in infinite systems, we studied the distribution of the correlation dimension D₂ and the nearest-neighbour level spacing distribution P(s) by paying attention to ALS. Results show that the influence of ALS to these distribution functions is limited. This is because the spatial distribution of amplitudes in tail regions of ALS exhibits multifractality as in the case of typical critical wave functions.     

Mapping of the anisotropic two-channel Anderson model onto a Fermi-Majorana biresonant level model

We establish the correspondence between an extended version of the two-channel Anderson model and a particular type of biresonant level model. For certain values of the parameters the new model becomes quadratic. We calculate in closed form the entropy and impurity occupation as functions of temperature and identify the different physical energy scales of the problem. We show how, as the temperature goes to zero, the model approaches a universal line of fixed points non-Fermi liquid in nature.     

On electron localization criteria in the Anderson model

The averaged Green function and the conductivity for the Anderson model are calculated for a simple cubic lattice within the CPA. The results are used to investigate the behaviour of the mobility edge on the basis of the two rather different approaches of the minimum metallic conductivity idea due to Mott and the Economou-Cohen localization theory.     

Self-consistent theory of localization for the Anderson model

The self-consistent theory of electron localization in a random system in the form proposed by Vollhardt and Wölfle is generalized for the analysis of localization in the Anderson model. We derive the general equations appropriate for the system with rather general form of the electronic spectrum. Explicit calculations are restricted to the lattices of cubic symmetry and use the effective mass approximation to obtain the final results. Anderson's critical ratio for the localization of all the electronic states in the tight-binding band is evaluated and found to be in surprisingly good agreement with the results of numerical analysis of localization in the Anderson model.     

Anderson localization in a two-dimensional random gap model

We study the properties of the spinor wavefunction in a strongly disordered environment on a two-dimensional lattice. By employing a transfer-matrix calculation we find that there is a transition from delocalized to localized states at a critical value of the disorder strength. We prove that there exists an Anderson localized phase with exponentially decaying correlations for sufficiently strong scattering. Our results indicate that suppressed backscattering is not sufficient to prevent Anderson localization of surface states in topological insulators.     

Solution of the two-channel Anderson impurity model: implications for the heavy fermion UBe13

We solve the two-channel Anderson impurity model using the Bethe-ansatz. We determine the ground state and derive the thermodynamics, obtaining the impurity entropy and specific heat over the full range of temperature. We show that the low-temperature physics is given by a line of fixed points describing a two-channel non-Fermi-liquid behavior in the integral valence regime associated with moment formation as well as in the mixed valence regime where no moment forms. We discuss the relevance for the theory of UBe13.     

Constructive proof of localization in the Anderson tight binding model

We prove that, for large disorder or near the band tails, the spectrum of the Anderson tight binding Hamiltonian with diagonal disorder consists exclusively of discrete eigenvalues. The corresponding eigenfunctions are exponentially well localized. These results hold in arbitrary dimension and with probability one. In one dimension, we recover the result that all states are localized for arbitrary energies and arbitrarily small disorder. Our techniques extend to other physical systems which exhibit localization phenomena, such as infinite systems of coupled harmonic oscillators, or random Schrödinger operators in the continuum.Work supported in part by National Science Foundations grant MCS-8108814 (A03).Work supported in part by National Science Foundation grant DMR 81-00417.     

A new proof of localization in the Anderson tight binding model

We give a new proof of exponential localization in the Anderson tight binding model which uses many ideas of the Frohlich, Martinelli, Scoppola and Spencer proof, but is technically simpler-particularly the probabilistic estimates.Partially supported by NSF Grant DMS 8702301     

Numerical Renormalization Group computation of temperature dependent specific heat for a two-channel Anderson model

The Numerical Renormalization Group (NRG) is applied to diagonalize a two-channel Anderson model describing a local magnetic impurity embedded in a fermionic bath. In spite of the difficulty in computing the specific heat using NRG, the interleaving discretization and multi-step iterative transformation virtually eliminate the numerical oscillations introduced by the logarithmic discretization of the conduction band. These allow to cover uniformly a large range of temperature, from the top of the band to a very small fraction of the bandwidth. This is relevant in describing, for instance, the presence of a low temperature Kondo resonance together with a high temperature Schottky peak, as well to cover Fermi and non-Fermi liquid regimes, like in the recent studied Ce_1−xLa_xNi₉Ge₄ family. We highlight the importance in describing the Schottky peak to define the number of degrees of freedom of the local levels, in order to correctly define the model to describe a given compound.     

Numerical study of localization in Anderson model for disordered systems

The localization properties in the Anderson model of two dimensional square lattices are investigated numerically. Pretty large lattices composed of 10⁴ (= 100 × 100) sites are dealt with, and the overall behaviors of the eigenvectors near the band center are directly examined. Fairly sharp transition and the exponentially decaying localized states are visualized.     

Anderson localization and non-linear sigma model with graded symmetry

A model of disordered single-particle systems is studied with regard to properties of the localized phase. Defined over a graded coset space, this model represents the correct non-perturbative extension of a non-linear sigma model introduced into localization theory by Schäfer and Wegner. An integral theorem is proven which allows us to change variables and execute the Grassmann integrations rather easily. In the localized phase, the invariant two-point functions are singular on the real axis. It is shown how to extract the singular contribution before evaluation of the functional integral. This is used to derive Efetov's solution of the Cayley tree model in a simple and transparent manner. Finally, a Monte Carlo algorithm is outlined which makes it possible to study Anderson localization in d > 2 dimensions.     

Chiral phase transition and Anderson localization in the instanton liquid model for QCD

We study the spectrum and eigenmodes of the QCD Dirac operator in a gauge background given by an instanton liquid model (ILM) at temperatures around the chiral phase transition. Generically we find the Dirac eigenvectors become more localized as the temperature is increased. At the chiral phase transition, both the low lying eigenmodes and the spectrum of the QCD Dirac operator undergo a transition to localization similar to the one observed in a disordered conductor. This suggests that Anderson localization is the fundamental mechanism driving the chiral phase transition. We also find an additional temperature dependent mobility edge (separating delocalized from localized eigenstates) in the bulk of the spectrum which moves toward lower eigenvalues as the temperature is increased. In both regions, the origin and the bulk, the transition to localization exhibits features of a 3D Anderson transition including multifractal eigenstates and spectral properties that are well described by critical statistics. Similar results are obtained in both the quenched and the unquenched case though the critical temperature in the unquenched case is lower. Finally we argue that our findings are not in principle restricted to the ILM approximation and may also be found in lattice simulations.     

Anderson localization from the replica formalism

We study Anderson localization in quasi-one-dimensional disordered wires within the framework of the replica sigma model. Applying a semiclassical approach (geodesic action plus Gaussian fluctuations) recently introduced within the context of supersymmetry by Lamacraft, Simons, and Zirnbauer, we compute the exact density of transmission matrix eigenvalues of superconducting wires (of symmetry class CI.) For the unitary class of metallic systems (class A) we are able to obtain the density function, save for its large transmission tail.     

Anderson localization in nonlocal nonlinear media

We theoretically and numerically investigate the effect of focusing and defocusing nonlinearities on Anderson localization in highly nonlocal media. A perturbative approach is developed to solve the nonlocal nonlinear Schr?dinger equation in the presence of a random potential, showing that nonlocality stabilizes Anderson states.