192 research outputs found
A Non-Hermitean Particle in a Disordered World
There has been much recent work on the spectrum of the random non-hermitean
Hamiltonian which models the physics of vortex line pinning in superconductors.
This note is loosely based on the talk I gave at the conference "New Directions
in Statistical Physics" held in Taipei, August 1997. We describe here new
results in spatial dimensions higher than one. We also give an expression for
the spectrum within the WKB approximation.Comment: latex file, 23 pages, 7 .ps figure
Characterisation of Anderson localisation using distributions
We examine the use of distributions in numerical treatments of Anderson
localisation and supply evidence that treating exponential localisation on
Bethe lattices recovers the overall picture known from hypercubic lattices in
3d.Comment: 2 pages, 2 figures, submitted to SCES'0
Stochastic Green's function approach to disordered systems
Based on distributions of local Green's functions we present a stochastic
approach to disordered systems. Specifically we address Anderson localisation
and cluster effects in binary alloys. Taking Anderson localisation of Holstein
polarons as an example we discuss how this stochastic approach can be used for
the investigation of interacting disordered systems.Comment: 12 pages, 7 figures, conference proceedings: Progress in
Nonequilibrium Green's Functions III, 22-26 August 2005, University of Kiel,
German
Spectral Curves of Non-Hermitean Hamiltonians
Recent analytical and numerical work have shown that the spectrum of the
random non-hermitean Hamiltonian on a ring which models the physics of vortex
line pinning in superconductors is one dimensional. In the maximally
non-hermitean limit, we give a simple "one-line" proof of this feature. We then
study the spectral curves for various distributions of the random site
energies. We find that a critical transition occurs when the average of the
logarithm of the random site energy squared vanishes. For a large class of
probability distributions of the site energies, we find that as the randomness
increases the energy at which the localization-delocalization transition occurs
increases, reaches a maximum, and then decreases. The Cauchy distribution
studied previously in the literature does not have this generic behavior. We
determine the critical value of the randomness at which "wings" first appear in
the energy spectrum. For distributions, such as Cauchy, with infinitely long
tails, we show that this critical value is infinitesimally above zero. We
determine the density of eigenvalues on the wings for any probability
distribution. We show that the localization length on the wings diverges
linearly as the energy approaches the energy at which the
localization-delocalization transition occurs. These results are all obtained
in the maximally non-hermitean limit but for a generic class of probability
distributions of the random site energies.Comment: 36 pages, 5 figures (.ps), LaTe
Spectra of Euclidean Random Matrices
We study the spectrum of a random matrix, whose elements depend on the
Euclidean distance between points randomly distributed in space. This problem
is widely studied in the context of the Instantaneous Normal Modes of fluids
and is particularly relevant at the glass transition. We introduce a systematic
study of this problem through its representation by a field theory. In this way
we can easily construct a high density expansion, which can be resummed
producing an approximation to the spectrum similar to the Coherent Potential
Approximation for disordered systems.Comment: 10 pages, 4 figure
Ergodicity breaking in a model showing many-body localization
We study the breaking of ergodicity measured in terms of return probability
in the evolution of a quantum state of a spin chain. In the non ergodic phase a
quantum state evolves in a much smaller fraction of the Hilbert space than
would be allowed by the conservation of extensive observables. By the anomalous
scaling of the participation ratios with system size we are led to consider the
distribution of the wave function coefficients, a standard observable in modern
studies of Anderson localization. We finally present a criterion for the
identification of the ergodicity breaking (many-body localization) transition
based on these distributions which is quite robust and well suited for
numerical investigations of a broad class of problems.Comment: 5 pages, 5 figures, final versio
Anderson transition on the Cayley tree as a traveling wave critical point for various probability distributions
For Anderson localization on the Cayley tree, we study the statistics of
various observables as a function of the disorder strength and the number
of generations. We first consider the Landauer transmission . In the
localized phase, its logarithm follows the traveling wave form where (i) the disorder-averaged value moves linearly
and the localization length
diverges as with (ii) the
variable is a fixed random variable with a power-law tail for large with , so that all
integer moments of are governed by rare events. In the delocalized phase,
the transmission remains a finite random variable as , and
we measure near criticality the essential singularity with . We then consider the
statistical properties of normalized eigenstates, in particular the entropy and
the Inverse Participation Ratios (I.P.R.). In the localized phase, the typical
entropy diverges as with , whereas it grows
linearly in in the delocalized phase. Finally for the I.P.R., we explain
how closely related variables propagate as traveling waves in the delocalized
phase. In conclusion, both the localized phase and the delocalized phase are
characterized by the traveling wave propagation of some probability
distributions, and the Anderson localization/delocalization transition then
corresponds to a traveling/non-traveling critical point. Moreover, our results
point towards the existence of several exponents at criticality.Comment: 28 pages, 21 figures, comments welcom
A single defect approximation for localized states on random lattices
Geometrical disorder is present in many physical situations giving rise to
eigenvalue problems. The simplest case of diffusion on a random lattice with
fluctuating site connectivities is studied analytically and by exact numerical
diagonalizations. Localization of eigenmodes is shown to be induced by
geometrical defects, that is sites with abnormally low or large connectivities.
We expose a ``single defect approximation'' (SDA) scheme founded on this
mechanism that provides an accurate quantitative description of both extended
and localized regions of the spectrum. We then present a systematic
diagrammatic expansion allowing to use SDA for finite-dimensional problems,
e.g. to determine the localized harmonic modes of amorphous media.Comment: final version as published, 6 pages, 1 ps-figur
Properties of low-lying states in a diffusive quantum dot and Fock-space localization
Motivated by an experiment by Sivan et al. (Europhys. Lett. 25, 605 (1994))
and by subsequent theoretical work on localization in Fock space, we study
numerically a hierarchical model for a finite many-body system of Fermions
moving in a disordered potential and coupled by a two-body interaction. We
focus attention on the low-lying states close to the Fermi energy. Both the
spreading width and the participation number depend smoothly on excitation
energy. This behavior is in keeping with naive expectations and does not
display Anderson localization. We show that the model reproduces essential
features of the experiment by Sivan et al.Comment: 4 pages, 3 figures, accepted for publication in Phys. Rev. Let
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