204 research outputs found
Delocalization in random polymer models
A random polymer model is a one-dimensional Jacobi matrix randomly composed
of two finite building blocks. If the two associated transfer matrices commute,
the corresponding energy is called critical. Such critical energies appear in
physical models, an example being the widely studied random dimer model. It is
proven that the Lyapunov exponent vanishes quadratically at a generic critical
energy and that the density of states is positive there. Large deviation
estimates around these asymptotics allow to prove optimal lower bounds on
quantum transport, showing that it is almost surely overdiffusive even though
the models are known to have pure-point spectrum with exponentially localized
eigenstates for almost every configuration of the polymers. Furthermore, the
level spacing is shown to be regular at the critical energy
Transport in the random Kronig-Penney model
The Kronig-Penney model with random Dirac potentials on the lattice \ZM has
critical energies at which the Lyapunov exponent vanishes and the density of
states has a van Hove singularity. This leads to a non-trivial quantum
diffusion even though the spectrum is known to be pure-point
Low density expansion for Lyapunov exponents
In some quasi-one-dimensional weakly disordered media, impurities are large
and rare rather than small and dense. For an Anderson model with a low density
of strong impurities, a perturbation theory in the impurity density is
developed for the Lyapunov exponent and the density of states. The Lyapunov
exponent grows linearly with the density. Anomalies of the Kappus-Wegner type
appear for all rational quasi-momenta even in lowest order perturbation theory
Spectral averaging techniques for Jacobi matrices
Spectral averaging techniques for one-dimensional discrete Schroedinger
operators are revisited and extended. In particular, simultaneous averaging
over several parameters is discussed. Special focus is put on proving lower
bounds on the density of the averaged spectral measures. These Wegner type
estimates are used to analyze stability properties for the spectral types of
Jacobi matrices under local perturbations
Random Dirac operators with time-reversal symmetry
Quasi-one-dimensional stochastic Dirac operators with an odd number of
channels, time reversal symmetry but otherwise efficiently coupled randomness
are shown to have one conducting channel and absolutely continuous spectrum of
multiplicity two. This follows by adapting the criteria of Guivarch-Raugi and
Goldsheid-Margulis to the analysis of random products of matrices in the group
SO, and then a version of Kotani theory for these operators. Absence of
singular spectrum can be shown by adapting an argument of Jaksic-Last if the
potential contains random Dirac peaks with absolutely continuous distribution.Comment: parts of introduction made more precise, corrections as follow-up on
referee report
Scattering theory for lattice operators in dimension
This paper analyzes the scattering theory for periodic tight-binding
Hamiltonians perturbed by a finite range impurity. The classical energy
gradient flow is used to construct a conjugate (or dilation) operator to the
unperturbed Hamiltonian. For dimension the wave operator is given by
an explicit formula in terms of this dilation operator, the free resolvent and
the perturbation. From this formula the scattering and time delay operators can
be read off. Using the index theorem approach, a Levinson theorem is proved
which also holds in presence of embedded eigenvalues and threshold
singularities.Comment: Minor errors and misprints corrected; new result on absense of
embedded eigenvalues for potential scattering; to appear in RM
Simultaneous quantization of edge and bulk Hall conductivity
The edge Hall conductivity is shown to be an integer multiple of
which is almost surely independent of the choice of the disordered
configuration. Its equality to the bulk Hall conductivity given by the
Kubo-Chern formula follows from K-theoretic arguments. This leads to
quantization of the Hall conductance for any redistribution of the current in
the sample. It is argued that in experiments at most a few percent of the total
current can be carried by edge states.Comment: 6 pages Latex, 1 figur
On the AC spectrum of one-dimensional random Schroedinger operators with matrix-valued potentials
We consider discrete one-dimensional random Schroedinger operators with
decaying matrix-valued, independent potentials. We show that if the l^2-norm of
this potential has finite expectation value with respect to the product measure
then almost surely the Schroedinger operator has an interval of purely
absolutely continuous (ac) spectrum. We apply this result to Schroedinger
operators on a strip. This work provides a new proof and generalizes a result
obtained by Delyon, Simon, and Souillard.Comment: (1 figure
Upper bounds on wavepacket spreading for random Jacobi matrices
A method is presented for proving upper bounds on the moments of the position
operator when the dynamics of quantum wavepackets is governed by a random
(possibly correlated) Jacobi matrix. As an application, one obtains sharp upper
bounds on the diffusion exponents for random polymer models, coinciding with
the lower bounds obtained in a prior work. The second application is an
elementary argument (not using multiscale analysis or the Aizenman-Molchanov
method) showing that under the condition of uniformly positive Lyapunov
exponents, the moments of the position operator grow at most logarithmically in
time.Comment: final version, to appear in CM
The scaling limit of the critical one-dimensional random Schrodinger operator
We consider two models of one-dimensional discrete random Schrodinger
operators (H_n \psi)_l ={\psi}_{l-1}+{\psi}_{l +1}+v_l {\psi}_l,
{\psi}_0={\psi}_{n+1}=0 in the cases v_k=\sigma {\omega}_k/\sqrt{n} and
v_k=\sigma {\omega}_k/ \sqrt{k}. Here {\omega}_k are independent random
variables with mean 0 and variance 1.
We show that the eigenvectors are delocalized and the transfer matrix
evolution has a scaling limit given by a stochastic differential equation. In
both cases, eigenvalues near a fixed bulk energy E have a point process limit.
We give bounds on the eigenvalue repulsion, large gap probability, identify the
limiting intensity and provide a central limit theorem.
In the second model, the limiting processes are the same as the point
processes obtained as the bulk scaling limits of the beta-ensembles of random
matrix theory. In the first model, the eigenvalue repulsion is much stronger.Comment: 36 pages, 2 figure
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