2,577 research outputs found

    On large-scale diagonalization techniques for the Anderson model of localization

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    We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely the computation of a few interior eigenvalues and their associated eigenvectors for large-scale sparse real and symmetric indefinite matrices of the Anderson model of localization. We compare the Lanczos algorithm in the 1987 implementation by Cullum and Willoughby with the shift-and-invert techniques in the implicitly restarted Lanczos method and in the Jacobiā€“Davidson method. Our preconditioning approaches for the shift-and-invert symmetric indefinite linear system are based on maximum weighted matchings and algebraic multilevel incomplete LDLT factorizations. These techniques can be seen as a complement to the alternative idea of using more complete pivoting techniques for the highly ill-conditioned symmetric indefinite Anderson matrices. We demonstrate the effectiveness and the numerical accuracy of these algorithms. Our numerical examples reveal that recent algebraic multilevel preconditioning solvers can accelerate the computation of a large-scale eigenvalue problem corresponding to the Anderson model of localization by several orders of magnitude

    Localization transition in one dimension using Wegner flow equations

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    The flow equation method was proposed by Wegner as a technique for studying interacting systems in one dimension. Here, we apply this method to a disordered one dimensional model with power-law decaying hoppings. This model presents a transition as function of the decaying exponent Ī±\alpha. We derive the flow equations, and the evolution of single-particle operators. The flow equation reveals the delocalized nature of the states for Ī±<1/2\alpha<1/2. Additionally, in the regime, Ī±>1/2\alpha>1/2, we present a strong-bond renormalization group structure based on iterating the three-site clusters, where we solve the flow equations perturbatively. This renormalization group approach allows us to probe the critical point (Ī±=1)\left(\alpha=1\right). This method correctly reproduces the critical level-spacing statistics, and the fractal dimensionality of the eigenfunctions.Comment: 19 pages, 16 figure

    Large Disorder Renormalization Group Study of the Anderson Model of Localization

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    We describe a large disorder renormalization group (LDRG) method for the Anderson model of localization in one dimension which decimates eigenstates based on the size of their wavefunctions rather than their energy. We show that our LDRG scheme flows to infinite disorder, and thus becomes asymptotically exact. We use it to obtain the disorder-averaged inverse participation ratio and density of states for the entire spectrum. A modified scheme is formulated for higher dimensions, which is found to be less efficient, but capable of improvement

    The forward approximation as a mean field approximation for the Anderson and Many Body Localization transitions

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    In this paper we analyze the predictions of the forward approximation in some models which exhibit an Anderson (single-) or many-body localized phase. This approximation, which consists in summing over the amplitudes of only the shortest paths in the locator expansion, is known to over-estimate the critical value of the disorder which determines the onset of the localized phase. Nevertheless, the results provided by the approximation become more and more accurate as the local coordination (dimensionality) of the graph, defined by the hopping matrix, is made larger. In this sense, the forward approximation can be regarded as a mean field theory for the Anderson transition in infinite dimensions. The sum can be efficiently computed using transfer matrix techniques, and the results are compared with the most precise exact diagonalization results available. For the Anderson problem, we find a critical value of the disorder which is 0.9%0.9\% off the most precise available numerical value already in 5 spatial dimensions, while for the many-body localized phase of the Heisenberg model with random fields the critical disorder hc=4.0Ā±0.3h_c=4.0\pm 0.3 is strikingly close to the most recent results obtained by exact diagonalization. In both cases we obtain a critical exponent Ī½=1\nu=1. In the Anderson case, the latter does not show dependence on the dimensionality, as it is common within mean field approximations. We discuss the relevance of the correlations between the shortest paths for both the single- and many-body problems, and comment on the connections of our results with the problem of directed polymers in random medium

    Emergent percolation length and localization in random elastic networks

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    We study, theoretically and numerically, a minimal model for phonons in a disordered system. For sufficient disorder, the vibrational modes of this classical system can become Anderson localized, yet this problem has received significantly less attention than its electronic counterpart. We find rich behavior in the localization properties of the phonons as a function of the density, frequency and the spatial dimension. We use a percolation analysis to argue for a Debye spectrum at low frequencies for dimensions higher than one, and for a localization/delocalization transition (at a critical frequency) above two dimensions. We show that in contrast to the behavior in electronic systems, the transition exists for arbitrarily large disorder, albeit with an exponentially small critical frequency. The structure of the modes reflects a divergent percolation length that arises from the disorder in the springs without being explicitly present in the definition of our model. Within the percolation approach we calculate the speed-of-sound of the delocalized modes (phonons), which we corroborate with numerics. We find the critical frequency of the localization transition at a given density, and find good agreement of these predictions with numerical results using a recursive Green function method adapted for this problem. The connection of our results to recent experiments on amorphous solids are discussed.Comment: accepted to PR

    Crossover of Level Statistics between Strong and Weak Localization in Two Dimensions

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    We investigate numerically the statistical properties of spectra of two-dimensional disordered systems by using the exact diagonalization and decimation method applied to the Anderson model. Statistics of spacings calculated for system sizes up to 1024 Ɨ\times 1024 lattice sites exhibits a crossover between Wigner and Poisson distributions. We perform a self-contained finite-size scaling analysis to find a single-valued one-parameter function Ī³(L/Ī¾)\gamma (L/\xi) which governs the crossover. The scaling parameter Ī¾(W)\xi(W) is deduced and compared with the localization length. Ī³(L/Ī¾)\gamma ( L/\xi) does {\em not} show critical behavior and has two asymptotic regimes corresponding to weakly and strongly localized states.Comment: 4 pages in revtex, 3 postscript figure
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