3,017 research outputs found
One-dimensional many-body entangled open quantum systems with tensor network methods
We present a collection of methods to simulate entangled dynamics of open
quantum systems governed by the Lindblad equation with tensor network methods.
Tensor network methods using matrix product states have been proven very useful
to simulate many-body quantum systems and have driven many innovations in
research. Since the matrix product state design is tailored for closed
one-dimensional systems governed by the Schr\"odinger equation, the next step
for many-body quantum dynamics is the simulation of open quantum systems. We
review the three dominant approaches to the simulation of open quantum systems
via the Lindblad master equation: quantum trajectories, matrix product density
operators, and locally purified tensor networks. Selected examples guide
possible applications of the methods and serve moreover as a benchmark between
the techniques. These examples include the finite temperature states of the
transverse quantum Ising model, the dynamics of an exciton traveling under the
influence of spontaneous emission and dephasing, and a double-well potential
simulated with the Bose-Hubbard model including dephasing. We analyze which
approach is favorable leading to the conclusion that a complete set of all
three methods is most beneficial, push- ing the limits of different scenarios.
The convergence studies using analytical results for macroscopic variables and
exact diagonalization methods as comparison, show, for example, that matrix
product density operators are favorable for the exciton problem in our study.
All three methods access the same library, i.e., the software package Open
Source Matrix Product States, allowing us to have a meaningful comparison
between the approaches based on the selected examples. For example, tensor
operations are accessed from the same subroutines and with the same
optimization eliminating one possible bias in a comparison of such numerical
methods.Comment: 24 pages, 8 figures. Small extension of time evolution section and
moving quantum simulators to introduction in comparison to v
The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems
We present a compendium of numerical simulation techniques, based on tensor
network methods, aiming to address problems of many-body quantum mechanics on a
classical computer. The core setting of this anthology are lattice problems in
low spatial dimension at finite size, a physical scenario where tensor network
methods, both Density Matrix Renormalization Group and beyond, have long proven
to be winning strategies. Here we explore in detail the numerical frameworks
and methods employed to deal with low-dimension physical setups, from a
computational physics perspective. We focus on symmetries and closed-system
simulations in arbitrary boundary conditions, while discussing the numerical
data structures and linear algebra manipulation routines involved, which form
the core libraries of any tensor network code. At a higher level, we put the
spotlight on loop-free network geometries, discussing their advantages, and
presenting in detail algorithms to simulate low-energy equilibrium states.
Accompanied by discussions of data structures, numerical techniques and
performance, this anthology serves as a programmer's companion, as well as a
self-contained introduction and review of the basic and selected advanced
concepts in tensor networks, including examples of their applications.Comment: 115 pages, 56 figure
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
Efficient Diagonalization of Kicked Quantum Systems
We show that the time evolution operator of kicked quantum systems, although
a full matrix of size NxN, can be diagonalized with the help of a new method
based on a suitable combination of fast Fourier transform and Lanczos algorithm
in just N^2 ln(N) operations. It allows the diagonalization of matrizes of
sizes up to N\approx 10^6 going far beyond the possibilities of standard
diagonalization techniques which need O(N^3) operations. We have applied this
method to the kicked Harper model revealing its intricate spectral properties.Comment: Text reorganized; part on the kicked Harper model extended. 13 pages
RevTex, 1 figur
Krylov-space approach to the equilibrium and the nonequilibrium single-particle Green's function
The zero-temperature single-particle Green's function of correlated fermion
models with moderately large Hilbert-space dimensions can be calculated by
means of Krylov-space techniques. The conventional Lanczos approach consists of
finding the ground state in a first step, followed by an approximation for the
resolvent of the Hamiltonian in a second step. We analyze the character of this
approximation and discuss a numerically exact variant of the Lanczos method
which is formulated in the time domain. This method is extended to get the
nonequilibrium single-particle Green's function defined on the
Keldysh-Matsubara contour in the complex time plane. The proposed method will
be important as an exact-diagonalization solver in the context of
self-consistent or variational cluster-embedding schemes. For the recently
developed nonequilibrium cluster-perturbation theory, we discuss the efficient
implementation and demonstrate the feasibility of the Krylov-based solver. The
dissipation of a strong local magnetic excitation into a non-interacting bath
is considered as an example for applications.Comment: 20 pages, 5 figures, v2 with minor corrections, JPCM in pres
Block Circulant and Toeplitz Structures in the Linearized Hartree–Fock Equation on Finite Lattices: Tensor Approach
This paper introduces and analyses the new grid-based tensor approach to
approximate solution of the elliptic eigenvalue problem for the 3D
lattice-structured systems. We consider the linearized Hartree-Fock equation
over a spatial lattice for both periodic and
non-periodic problem setting, discretized in the localized Gaussian-type
orbitals basis. In the periodic case, the Galerkin system matrix obeys a
three-level block-circulant structure that allows the FFT-based
diagonalization, while for the finite extended systems in a box (Dirichlet
boundary conditions) we arrive at the perturbed block-Toeplitz representation
providing fast matrix-vector multiplication and low storage size. The proposed
grid-based tensor techniques manifest the twofold benefits: (a) the entries of
the Fock matrix are computed by 1D operations using low-rank tensors
represented on a 3D grid, (b) in the periodic case the low-rank tensor
structure in the diagonal blocks of the Fock matrix in the Fourier space
reduces the conventional 3D FFT to the product of 1D FFTs. Lattice type systems
in a box with Dirichlet boundary conditions are treated numerically by our
previous tensor solver for single molecules, which makes possible calculations
on rather large lattices due to reduced numerical
cost for 3D problems. The numerical simulations for both box-type and periodic
lattice chain in a 3D rectangular "tube" with up to
several hundred confirm the theoretical complexity bounds for the
block-structured eigenvalue solvers in the limit of large .Comment: 30 pages, 12 figures. arXiv admin note: substantial text overlap with
arXiv:1408.383
Density Matrix Renormalization Group for Dummies
We describe the Density Matrix Renormalization Group algorithms for time
dependent and time independent Hamiltonians. This paper is a brief but
comprehensive introduction to the subject for anyone willing to enter in the
field or write the program source code from scratch.Comment: 29 pages, 9 figures. Published version. An open source version of the
code can be found at http://qti.sns.it/dmrg/phome.htm
A Sparse SCF algorithm and its parallel implementation: Application to DFTB
We present an algorithm and its parallel implementation for solving a self
consistent problem as encountered in Hartree Fock or Density Functional Theory.
The algorithm takes advantage of the sparsity of matrices through the use of
local molecular orbitals. The implementation allows to exploit efficiently
modern symmetric multiprocessing (SMP) computer architectures. As a first
application, the algorithm is used within the density functional based tight
binding method, for which most of the computational time is spent in the linear
algebra routines (diagonalization of the Fock/Kohn-Sham matrix). We show that
with this algorithm (i) single point calculations on very large systems
(millions of atoms) can be performed on large SMP machines (ii) calculations
involving intermediate size systems (1~000--100~000 atoms) are also strongly
accelerated and can run efficiently on standard servers (iii) the error on the
total energy due to the use of a cut-off in the molecular orbital coefficients
can be controlled such that it remains smaller than the SCF convergence
criterion.Comment: 13 pages, 11 figure
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