1,144 research outputs found
The density-matrix renormalization group
The density-matrix renormalization group (DMRG) is a numerical algorithm for
the efficient truncation of the Hilbert space of low-dimensional strongly
correlated quantum systems based on a rather general decimation prescription.
This algorithm has achieved unprecedented precision in the description of
one-dimensional quantum systems. It has therefore quickly acquired the status
of method of choice for numerical studies of one-dimensional quantum systems.
Its applications to the calculation of static, dynamic and thermodynamic
quantities in such systems are reviewed. The potential of DMRG applications in
the fields of two-dimensional quantum systems, quantum chemistry,
three-dimensional small grains, nuclear physics, equilibrium and
non-equilibrium statistical physics, and time-dependent phenomena is discussed.
This review also considers the theoretical foundations of the method, examining
its relationship to matrix-product states and the quantum information content
of the density matrices generated by DMRG.Comment: accepted by Rev. Mod. Phys. in July 2004; scheduled to appear in the
January 2005 issu
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
Benefits from using mixed precision computations in the ELPA-AEO and ESSEX-II eigensolver projects
We first briefly report on the status and recent achievements of the ELPA-AEO
(Eigenvalue Solvers for Petaflop Applications - Algorithmic Extensions and
Optimizations) and ESSEX II (Equipping Sparse Solvers for Exascale) projects.
In both collaboratory efforts, scientists from the application areas,
mathematicians, and computer scientists work together to develop and make
available efficient highly parallel methods for the solution of eigenvalue
problems. Then we focus on a topic addressed in both projects, the use of mixed
precision computations to enhance efficiency. We give a more detailed
description of our approaches for benefiting from either lower or higher
precision in three selected contexts and of the results thus obtained
The Density Matrix Renormalization Group for finite Fermi systems
The Density Matrix Renormalization Group (DMRG) was introduced by Steven
White in 1992 as a method for accurately describing the properties of
one-dimensional quantum lattices. The method, as originally introduced, was
based on the iterative inclusion of sites on a real-space lattice. Based on its
enormous success in that domain, it was subsequently proposed that the DMRG
could be modified for use on finite Fermi systems, through the replacement of
real-space lattice sites by an appropriately ordered set of single-particle
levels. Since then, there has been an enormous amount of work on the subject,
ranging from efforts to clarify the optimal means of implementing the algorithm
to extensive applications in a variety of fields. In this article, we review
these recent developments. Following a description of the real-space DMRG
method, we discuss the key steps that were undertaken to modify it for use on
finite Fermi systems and then describe its applications to Quantum Chemistry,
ultrasmall superconducting grains, finite nuclei and two-dimensional electron
systems. We also describe a recent development which permits symmetries to be
taken into account consistently throughout the DMRG algorithm. We close with an
outlook for future applications of the method.Comment: 48 pages, 17 figures Corrections made to equation 19 and table
Rigorous numerical approaches in electronic structure theory
Electronic structure theory concerns the description of molecular properties according to the postulates of quantum mechanics. For practical purposes, this is realized entirely through numerical computation, the scope of which is constrained by computational costs that increases rapidly with the size of the system.
The significant progress made in this field over the past decades have been facilitated in part by the willingness of chemists to forego some mathematical rigour in exchange for greater efficiency. While such compromises allow large systems to be computed feasibly, there are lingering concerns over the impact that these compromises have on the quality of the results that are produced. This research is motivated by two key issues that contribute to this loss of quality, namely i) the numerical errors accumulated due to the use of finite precision arithmetic and the application of numerical approximations, and ii) the reliance on iterative methods that are not guaranteed to converge to the correct solution.
Taking the above issues in consideration, the aim of this thesis is to explore ways to perform electronic structure calculations with greater mathematical rigour, through the application of rigorous numerical methods. Of which, we focus in particular on methods based on interval analysis and deterministic global optimization. The Hartree-Fock electronic structure method will be used as the subject of this study due to its ubiquity within this domain.
We outline an approach for placing rigorous bounds on numerical error in Hartree-Fock computations. This is achieved through the application of interval analysis techniques, which are able to rigorously bound and propagate quantities affected by numerical errors. Using this approach, we implement a program called Interval Hartree-Fock. Given a closed-shell system and the current electronic state, this program is able to compute rigorous error bounds on quantities including i) the total energy, ii) molecular orbital energies, iii) molecular orbital coefficients, and iv) derived electronic properties.
Interval Hartree-Fock is adapted as an error analysis tool for studying the impact of numerical error in Hartree-Fock computations. It is used to investigate the effect of input related factors such as system size and basis set types on the numerical accuracy of the Hartree-Fock total energy. Consideration is also given to the impact of various algorithm design decisions. Examples include the application of different integral screening thresholds, the variation between single and double precision arithmetic in two-electron integral evaluation, and the adjustment of interpolation table granularity. These factors are relevant to both the usage of conventional Hartree-Fock code, and the development of Hartree-Fock code optimized for novel computing devices such as graphics processing units.
We then present an approach for solving the Hartree-Fock equations to within a guaranteed margin of error. This is achieved by treating the Hartree-Fock equations as a non-convex global optimization problem, which is then solved using deterministic global optimization. The main contribution of this work is the development of algorithms for handling quantum chemistry specific expressions such as the one and two-electron integrals within the deterministic global optimization framework. This approach was implemented as an extension to an existing open source solver.
Proof of concept calculations are performed for a variety of problems within Hartree-Fock theory, including those in i) point energy calculation, ii) geometry optimization, iii) basis set optimization, and iv) excited state calculation. Performance analyses of these calculations are also presented and discussed
A fast iterative algorithm for near-diagonal eigenvalue problems
We introduce a novel iterative eigenvalue algorithm for near-diagonal
matrices termed iterative perturbative theory (IPT). Built upon a
"perturbative" partitioning of the matrix into diagonal and off-diagonal parts,
IPT computes one or all eigenpairs with a complexity per iteration of one
matrix-vector or one matrix-matrix multiplication respectively. Thanks to the
high parallelism of these basic linear algebra operations, we obtain excellent
performance on multi-core processors and GPUs, with large speed-ups over
standard methods (up to x with respect to LAPACK and ARPACK). For
matrices which are not close to being diagonal but have well-separated
eigenvalues, IPT can be be used to refine low-precision eigenpairs obtained by
other methods. We give sufficient conditions for linear convergence and
demonstrate performance on dense and sparse test matrices. In a real-world
application from quantum chemistry, we find that IPT performs similarly to the
Davidson algorithm.Comment: Based on arXiv:2002.1287
The numerical renormalization group method for quantum impurity systems
In the beginning of the 1970's, Wilson developed the concept of a fully
non-perturbative renormalization group transformation. Applied to the Kondo
problem, this numerical renormalization group method (NRG) gave for the first
time the full crossover from the high-temperature phase of a free spin to the
low-temperature phase of a completely screened spin. The NRG has been later
generalized to a variety of quantum impurity problems. The purpose of this
review is to give a brief introduction to the NRG method including some
guidelines of how to calculate physical quantities, and to survey the
development of the NRG method and its various applications over the last 30
years. These applications include variants of the original Kondo problem such
as the non-Fermi liquid behavior in the two-channel Kondo model, dissipative
quantum systems such as the spin-boson model, and lattice systems in the
framework of the dynamical mean field theory.Comment: 55 pages, 27 figures, submitted to Rev. Mod. Phy
Efficiency of different numerical methods for solving Redfield equations
The numerical efficiency of different schemes for solving the Liouville-von
Neumann equation within multilevel Redfield theory has been studied. Among the
tested algorithms are the well-known Runge-Kutta scheme in two different
implementations as well as methods especially developed for time propagation:
the Short Iterative Arnoldi, Chebyshev and Newtonian propagators. In addition,
an implementation of a symplectic integrator has been studied. For a simple
example of a two-center electron transfer system we discuss some aspects of the
efficiency of these methods to integrate the equations of motion. Overall for
time-independent potentials the Newtonian method is recommended. For
time-dependent potentials implementations of the Runge-Kutta algorithm are very
efficient
Shift-invert diagonalization of large many-body localizing spin chains
We provide a pedagogical review on the calculation of highly excited
eigenstates of disordered interacting quantum systems which can undergo a
many-body localization (MBL) transition, using shift-invert exact
diagonalization. We also provide an example code at
https://bitbucket.org/dluitz/sinvert_mbl/. Through a detailed analysis of the
simulational parameters of the random field Heisenberg spin chain, we provide a
practical guide on how to perform efficient computations. We present data for
mid-spectrum eigenstates of spin chains of sizes up to . This work is
also geared towards readers with interest in efficiency of parallel sparse
linear algebra techniques that will find a challenging application in the MBL
problem
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