25,311 research outputs found
Diagonalization- and Numerical Renormalization-Group-Based Methods for Interacting Quantum Systems
In these lecture notes, we present a pedagogical review of a number of
related {\it numerically exact} approaches to quantum many-body problems. In
particular, we focus on methods based on the exact diagonalization of the
Hamiltonian matrix and on methods extending exact diagonalization using
renormalization group ideas, i.e., Wilson's Numerical Renormalization Group
(NRG) and White's Density Matrix Renormalization Group (DMRG). These methods
are standard tools for the investigation of a variety of interacting quantum
systems, especially low-dimensional quantum lattice models. We also survey
extensions to the methods to calculate properties such as dynamical quantities
and behavior at finite temperature, and discuss generalizations of the DMRG
method to a wider variety of systems, such as classical models and quantum
chemical problems. Finally, we briefly review some recent developments for
obtaining a more general formulation of the DMRG in the context of matrix
product states as well as recent progress in calculating the time evolution of
quantum systems using the DMRG and the relationship of the foundations of the
method with quantum information theory.Comment: 51 pages; lecture notes on numerically exact methods. Pedagogical
review appearing in the proceedings of the "IX. Training Course in the
Physics of Correlated Electron Systems and High-Tc Superconductors", Vietri
sul Mare (Salerno, Italy, October 2004
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
The Lagrange and Markov spectra from the dynamical point of view
This text grew out of my lecture notes for a 4-hours minicourse delivered on
October 17 \& 19, 2016 during the research school "Applications of Ergodic
Theory in Number Theory" -- an activity related to the Jean-Molet Chair project
of Mariusz Lema\'nczyk and S\'ebastien Ferenczi -- realized at CIRM, Marseille,
France. The subject of this text is the same of my minicourse, namely, the
structure of the so-called Lagrange and Markov spectra (with an special
emphasis on a recent theorem of C. G. Moreira).Comment: 27 pages, 6 figures. Survey articl
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