2,577 research outputs found
On large-scale diagonalization techniques for the Anderson model of localization
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 JacobiāDavidson 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
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 . We derive
the flow equations, and the evolution of single-particle operators. The flow
equation reveals the delocalized nature of the states for .
Additionally, in the regime, , 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 . 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
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
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
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 is strikingly close
to the most recent results obtained by exact diagonalization. In both cases we
obtain a critical exponent . 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
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
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 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
which governs the crossover. The scaling parameter is
deduced and compared with the localization length. 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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