65 research outputs found
Interaction induced delocalization of two particles: large system size calculations and dependence on interaction strength
The localization length of two interacting particles in a
one-dimensional disordered system is studied for very large system sizes by two
efficient and accurate variants of the Green function method. The numerical
results (at the band center) can be well described by the functional form
where is the one-particle localization length and
the coefficient depends on the strength of
the on-site Hubbard interaction. The Breit-Wigner width or equivalently the
(inverse) life time of non-interacting pair states is analytically calculated
for small disorder and taking into account the energy dependence of the
one-particle localization length. This provides a consistent theoretical
explanation of the numerically found -dependence of .Comment: 8 pages, 5 figures, LaTeX, EPJ macro package, submitted to the
European Physical Journal
Eigenfunction structure and scaling of two interacting particles in the one-dimensional Anderson model
The localization properties of eigenfunctions for two interacting particles
in the one-dimensional Anderson model are studied for system sizes up to
sites corresponding to a Hilbert space of dimension
using the Green function Arnoldi method. The eigenfunction structure is
illustrated in position, momentum and energy representation, the latter
corresponding to an expansion in non-interacting product eigenfunctions.
Different types of localization lengths are computed for parameter ranges in
system size, disorder and interaction strengths inaccessible until now. We
confirm that one-parameter scaling theory can be successfully applied provided
that the condition of being significantly larger than the one-particle
localization length is verified. The enhancement effect of the
two-particle localization length behaving as is clearly
confirmed for a certain quite large interval of optimal interactions strengths.
Further new results for the interaction dependence in a very large interval, an
energy value outside the band center, and different interaction ranges are
obtained.Comment: 26 pages, 19 png and pdf figures, high quality gif files for panels
of figures 1-4 are available at
http://www.quantware.ups-tlse.fr/QWLIB/tipdisorder1d, final published version
with minor corrections/revisions, addition of Journal reference and DO
Localization and absence of Breit-Wigner form for Cauchy random band matrices
We analytically calculate the local density of states for Cauchy random band
matrices with strongly fluctuating diagonal elements. The Breit-Wigner form for
ordinary band matrices is replaced by a Levy distribution of index
and the characteristic energy scale is strongly enhanced as compared
to the Breit-Wigner width. The unperturbed eigenstates decay according to the
non-exponential law . We analytically determine
the localization length by a new method to derive the supersymmetric non-linear
model for this type of band matrices.Comment: 4 pages, 1 figur
Poincar\'e recurrences and Ulam method for the Chirikov standard map
We study numerically the statistics of Poincar\'e recurrences for the
Chirikov standard map and the separatrix map at parameters with a critical
golden invariant curve. The properties of recurrences are analyzed with the
help of a generalized Ulam method. This method allows to construct the
corresponding Ulam matrix whose spectrum and eigenstates are analyzed by the
powerful Arnoldi method. We also develop a new survival Monte Carlo method
which allows us to study recurrences on times changing by ten orders of
magnitude. We show that the recurrences at long times are determined by
trajectory sticking in a vicinity of the critical golden curve and secondary
resonance structures. The values of Poincar\'e exponents of recurrences are
determined for the two maps studied. We also discuss the localization
properties of eigenstates of the Ulam matrix and their relation with the
Poincar\'e recurrences.Comment: 11 pages, 14 figures, high resolution figures and video mpeg files
available at: http://www.quantware.ups-tlse.fr/QWLIB/ulammethod
Freed by interaction kinetic states in the Harper model
We study the problem of two interacting particles in a one-dimensional
quasiperiodic lattice of the Harper model. We show that a short or long range
interaction between particles leads to emergence of delocalized pairs in the
non-interacting localized phase. The properties of these Freed by Interaction
Kinetic States (FIKS) are analyzed numerically including the advanced Arnoldi
method. We find that the number of sites populated by FIKS pairs grows
algebraically with the system size with the maximal exponent , up to a
largest lattice size reached in our numerical simulations, thus
corresponding to a complete delocalization of pairs. For delocalized FIKS pairs
the spectral properties of such quasiperiodic operators represent a deep
mathematical problem. We argue that FIKS pairs can be detected in the framework
of recent cold atom experiments [M.~Schreiber {\it et al.} Science {\bf 349},
842 (2015)] by a simple setup modification. We also discuss possible
implications of FIKS pairs for electron transport in the regime of
charge-density wave and high superconductivity.Comment: 26 pages, 21 pdf and png figures, additional data and high quality
figures are available at http://www.quantware.ups-tlse.fr/QWLIB/fikspairs/ ,
parts of sections 2 and 3 moved to appendices, manuscript accepted for EPJ
Dynamical decoherence of a qubit coupled to a quantum dot or the SYK black hole
We study the dynamical decoherence of a qubit weakly coupled to a two-body
random interaction model (TBRIM) describing a quantum dot of interacting
fermions or the Sachdev-Ye-Kitaev (SYK) black hole model. We determine the
rates of qubit relaxation and dephasing for regimes of dynamical thermalization
of the quantum dot or of quantum chaos in the SYK model. These rates are found
to correspond to the Fermi golden rule and quantum Zeno regimes depending on
the qubit-fermion coupling strength. An unusual regime is found where these
rates are practically independent of TBRIM parameters. We push forward an
analogy between TBRIM and quantum small-world networks with an explosive
spreading over exponentially large number of states in a finite time being
similar to six degrees of separation in small-world social networks. We find
that the SYK model has approximately two-three degrees of separation.Comment: 17 pages, 15 pdf-figure
Spectral properties of Google matrix of Wikipedia and other networks
We study the properties of eigenvalues and eigenvectors of the Google matrix
of the Wikipedia articles hyperlink network and other real networks. With the
help of the Arnoldi method we analyze the distribution of eigenvalues in the
complex plane and show that eigenstates with significant eigenvalue modulus are
located on well defined network communities. We also show that the correlator
between PageRank and CheiRank vectors distinguishes different organizations of
information flow on BBC and Le Monde web sites.Comment: 10 pages, 9 figure
Google matrix analysis of directed networks
In past ten years, modern societies developed enormous communication and
social networks. Their classification and information retrieval processing
become a formidable task for the society. Due to the rapid growth of World Wide
Web, social and communication networks, new mathematical methods have been
invented to characterize the properties of these networks on a more detailed
and precise level. Various search engines are essentially using such methods.
It is highly important to develop new tools to classify and rank enormous
amount of network information in a way adapted to internal network structures
and characteristics. This review describes the Google matrix analysis of
directed complex networks demonstrating its efficiency on various examples
including World Wide Web, Wikipedia, software architecture, world trade, social
and citation networks, brain neural networks, DNA sequences and Ulam networks.
The analytical and numerical matrix methods used in this analysis originate
from the fields of Markov chains, quantum chaos and Random Matrix theory.Comment: 56 pages, 58 figures. Missed link added in network example of Fig3
Google matrix of the citation network of Physical Review
We study the statistical properties of spectrum and eigenstates of the Google
matrix of the citation network of Physical Review for the period 1893 - 2009.
The main fraction of complex eigenvalues with largest modulus is determined
numerically by different methods based on high precision computations with up
to binary digits that allows to resolve hard numerical problems for
small eigenvalues. The nearly nilpotent matrix structure allows to obtain a
semi-analytical computation of eigenvalues. We find that the spectrum is
characterized by the fractal Weyl law with a fractal dimension .
It is found that the majority of eigenvectors are located in a localized phase.
The statistical distribution of articles in the PageRank-CheiRank plane is
established providing a better understanding of information flows on the
network. The concept of ImpactRank is proposed to determine an influence domain
of a given article. We also discuss the properties of random matrix models of
Perron-Frobenius operators.Comment: 25 pages. 17 figures. Published in Phys. Rev.
- âŠ