30 research outputs found
Multi-Particle Anderson Localisation: Induction on the Number of Particles
This paper is a follow-up of our recent papers \cite{CS08} and \cite{CS09}
covering the two-particle Anderson model. Here we establish the phenomenon of
Anderson localisation for a quantum -particle system on a lattice
with short-range interaction and in presence of an IID external potential with
sufficiently regular marginal cumulative distribution function (CDF). Our main
method is an adaptation of the multi-scale analysis (MSA; cf. \cite{FS},
\cite{FMSS}, \cite{DK}) to multi-particle systems, in combination with an
induction on the number of particles, as was proposed in our earlier manuscript
\cite{CS07}. Similar results have been recently obtained in an independent work
by Aizenman and Warzel \cite{AW08}: they proposed an extension of the
Fractional-Moment Method (FMM) developed earlier for single-particle models in
\cite{AM93} and \cite{ASFH01} (see also references therein) which is also
combined with an induction on the number of particles.
An important role in our proof is played by a variant of Stollmann's
eigenvalue concentration bound (cf. \cite{St00}). This result, as was proved
earlier in \cite{C08}, admits a straightforward extension covering the case of
multi-particle systems with correlated external random potentials: a subject of
our future work. We also stress that the scheme of our proof is \textit{not}
specific to lattice systems, since our main method, the MSA, admits a
continuous version. A proof of multi-particle Anderson localization in
continuous interacting systems with various types of external random potentials
will be published in a separate papers
On the Joint Distribution of Energy Levels of Random Schroedinger Operators
We consider operators with random potentials on graphs, such as the lattice
version of the random Schroedinger operator. The main result is a general bound
on the probabilities of simultaneous occurrence of eigenvalues in specified
distinct intervals, with the corresponding eigenfunctions being separately
localized within prescribed regions. The bound generalizes the Wegner estimate
on the density of states. The analysis proceeds through a new multiparameter
spectral averaging principle
Quantum harmonic oscillator systems with disorder
We study many-body properties of quantum harmonic oscillator lattices with
disorder. A sufficient condition for dynamical localization, expressed as a
zero-velocity Lieb-Robinson bound, is formulated in terms of the decay of the
eigenfunction correlators for an effective one-particle Hamiltonian. We show
how state-of-the-art techniques for proving Anderson localization can be used
to prove that these properties hold in a number of standard models. We also
derive bounds on the static and dynamic correlation functions at both zero and
positive temperature in terms of one-particle eigenfunction correlators. In
particular, we show that static correlations decay exponentially fast if the
corresponding effective one-particle Hamiltonian exhibits localization at low
energies, regardless of whether there is a gap in the spectrum above the ground
state or not. Our results apply to finite as well as to infinite oscillator
systems. The eigenfunction correlators that appear are more general than those
previously studied in the literature. In particular, we must allow for
functions of the Hamiltonian that have a singularity at the bottom of the
spectrum. We prove exponential bounds for such correlators for some of the
standard models
Localization Bounds for Multiparticle Systems
We consider the spectral and dynamical properties of quantum systems of
particles on the lattice , of arbitrary dimension, with a Hamiltonian
which in addition to the kinetic term includes a random potential with iid
values at the lattice sites and a finite-range interaction. Two basic
parameters of the model are the strength of the disorder and the strength of
the interparticle interaction. It is established here that for all there
are regimes of high disorder, and/or weak enough interactions, for which the
system exhibits spectral and dynamical localization. The localization is
expressed through bounds on the transition amplitudes, which are uniform in
time and decay exponentially in the Hausdorff distance in the configuration
space. The results are derived through the analysis of fractional moments of
the -particle Green function, and related bounds on the eigenfunction
correlators
A Nonperturbative Eliasson's Reducibility Theorem
This paper is concerned with discrete, one-dimensional Schr\"odinger
operators with real analytic potentials and one Diophantine frequency. Using
localization and duality we show that almost every point in the spectrum admits
a quasi-periodic Bloch wave if the potential is smaller than a certain constant
which does not depend on the precise Diophantine conditions. The associated
first-order system, a quasi-periodic skew-product, is shown to be reducible for
almost all values of the energy. This is a partial nonperturbative
generalization of a reducibility theorem by Eliasson. We also extend
nonperturbatively the genericity of Cantor spectrum for these Schr\"odinger
operators. Finally we prove that in our setting, Cantor spectrum implies the
existence of a -set of energies whose Schr\"odinger cocycle is not
reducible to constant coefficients
Two interacting Hofstadter butterflies
The problem of two interacting particles in a quasiperiodic potential is
addressed. Using analytical and numerical methods, we explore the spectral
properties and eigenstates structure from the weak to the strong interaction
case. More precisely, a semiclassical approach based on non commutative
geometry techniques permits to understand the intricate structure of such a
spectrum. An interaction induced localization effect is furthermore emphasized.
We discuss the application of our results on a two-dimensional model of two
particles in a uniform magnetic field with on-site interaction.Comment: revtex, 12 pages, 11 figure
Disorder-assisted error correction in Majorana chains
It was recently realized that quenched disorder may enhance the reliability
of topological qubits by reducing the mobility of anyons at zero temperature.
Here we compute storage times with and without disorder for quantum chains with
unpaired Majorana fermions - the simplest toy model of a quantum memory.
Disorder takes the form of a random site-dependent chemical potential. The
corresponding one-particle problem is a one-dimensional Anderson model with
disorder in the hopping amplitudes. We focus on the zero-temperature storage of
a qubit encoded in the ground state of the Majorana chain. Storage and
retrieval are modeled by a unitary evolution under the memory Hamiltonian with
an unknown weak perturbation followed by an error-correction step. Assuming
dynamical localization of the one-particle problem, we show that the storage
time grows exponentially with the system size. We give supporting evidence for
the required localization property by estimating Lyapunov exponents of the
one-particle eigenfunctions. We also simulate the storage process for chains
with a few hundred sites. Our numerical results indicate that in the absence of
disorder, the storage time grows only as a logarithm of the system size. We
provide numerical evidence for the beneficial effect of disorder on storage
times and show that suitably chosen pseudorandom potentials can outperform
random ones.Comment: 50 pages, 7 figure
Anderson localisation in stationary ensembles of quasiperiodic operators
An ensemble of quasi-periodic discrete Schro ̈dinger operators with an arbitrary number of basic frequencies is considered, in a lattice of arbitrary dimension, in which the hull function is a realisation of a stationary Gaussian process on the torus. We show that, for almost every element of the ensemble, the quasi-periodic operator boasts Anderson localization with simple pure point spectrum at strong coupling. One of the ingredients of the proof is a new lower bound on the interpolation error for stationary Gaussian processes on the torus (also known as local non-determinism)