Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Brazilian Journal of Physics

A simple way to avoid metastable configurations in the density-matrix renormalization-group algorithms

J. C. Xavier

Published Sep 1, 2009• 4 citations
Publisher

Abstract

We use the spin-1 Heisenberg chain with periodic boundary conditions to ilustrate that the systems get stuck in metastable configurations only when the density-matrix renormalization-group algorithm start with small number of states m. We also show that the convergence of the energies have a huge improvement if we start the algorithm with a large number of states m.

DOI
10.1590/s0103-97332009000500016

Related papers

Density matrix renormalization group algorithms with a single center site

Steven R. White

We develop a correction to the density matrix used in density matrix renormalization group calculations to take into account the incompleteness of the environment block. The correction allows successful calculations using only a single site in the center of the system, rather than the standard two sites, improving typical computation times by a factor of two to four. In addition, in many cases where the ordinary density matrix renormalization group method can get stuck in metastable configurations, the correction eliminates the sticking. We test the method on the Heisenberg $S=1$ chain.

arXiv:cond-mat/0508709•Physical Review B•Nov 17, 2005•239 citations

Density-matrix renormalization-group algorithm for quantum lattice systems with a large number of states per site

Robert J. Bursill

A variant of White's density-matrix renormalization-group scheme, which is designed to compute low-lying energies of one-dimensional quantum lattice models with a large number of degrees of freedom per site is described. The method is tested on two exactly solvable models---the spin-$\frac{1}{2}$ antiferromagnetic Heisenberg chain and a dimerized $\mathrm{XY}$ spin chain. To illustrate the potential of the method, it is applied to a model of spins interacting with quantum phonons. It is shown that the method accurately resolves a number of energy gaps on periodic rings that are sufficiently large to afford an accurate investigation of critical properties via the use of finite-size scaling theory.

arXiv:cond-mat/9812349•Physical review. B, Condensed matter•Jul 15, 1999•31 citations

Density matrix renormalization group (DMRG) for cyclic and centrosymmetric linear chains

Manoranjan Kumar, Dayasindhu Dey, Aslam Parvej +2 more

The density matrix renormalization group (DMRG) method generates the low-energy states of linear systems of $N$ sites with a few degrees of freedom at each site by starting with a small system and adding sites step by step while keeping constant the dimension of the truncated Hilbert space. DMRG algorithms are adapted to open chains with inversion symmetry at the central site, to cyclic chains and to weakly coupled chains. Physical properties rather than energy accuracy is the motivation. The algorithms are applied to the edge states of linear Heisenberg antiferromagnets with spin $S \ge 1$ and to the quantum phases of a frustrated spin-1/2 chain with exchange between first and second neighbors. The algorithms are found to be accurate for extended Hubbard and related 1D models with charge and spin degrees of freedom.

arXiv:1610.00998•Oct 4, 2016•0 citations

Wave-function initialization in an infinite-chain density-matrix renormalization-group algorithm

Liqun Sun, Jun Zhang, Shaojin Qin +1 more

An improved trial-wave-function initialization approach is suggested in an infinite-chain density-matrix renormalization-group (DMRG) method. When we remove the sign freedom of the retained basis in a DMRG algorithm, an optimal trial wave function for a chain can be obtained from previous wave functions for chains with shorter lengths in DMRG. Such an optimal trial wave function will reduce the iteration number of Lanczos-style approaches in calculating the eigen-wave-functions. The possibilities of converging to other eigenenergy levels are also eliminated. A one-dimensional Heisenberg chain is used in this paper to illustrate our practice.

Physical review. B, Condensed matter•Mar 20, 2002•3 citations

Thermodynamic Limit of Density Matrix Renormalization

Stellan S. Ostlund, Stefan Rommer

The density matrix renormalization group discovered by White is investigated. In the case where renormalization eventually converges to a fixed point we show that quantum states in the thermodynamic limit with periodic boundary conditions can be simply represented by a ``matrix product ground state'' with a natural description of Bloch states of elementary excitations. We then observe that these states can be rederived through a simple variational ansatz making no reference to a renormalization construction. The method is tested on the spin-1 Heisenberg model.

arXiv:cond-mat/9503107•Physical Review Letters•Nov 6, 1995•1,077 citations

Obtaining the two-body density matrix in the density matrix renormalization group method

Dominika Zgid, Marcel Nooijen

We present an approach that allows to produce the two-body density matrix during the density matrix renormalization group (DMRG) run without an additional increase in the current disk and memory requirements. The computational cost of producing the two-body density matrix is proportional to O(M3k2+M2k4). The method is based on the assumption that different elements of the two-body density matrix can be calculated during different steps of a sweep. Hence, it is desirable that the wave function at the convergence does not change during a sweep. We discuss the theoretical structure of the wave function ansatz used in DMRG, concluding that during the one-site DMRG procedure, the energy and the wave function are converging monotonically at every step of the sweep. Thus, the one-site algorithm provides an opportunity to obtain the two-body density matrix free from the N-representability problem. We explain the problem of local minima that may be encountered in the DMRG calculations. We discuss theoretically why and when the one- and two-site DMRG procedures may get stuck in a metastable solution, and we list practical solutions helping the minimization to avoid the local minima.

The Journal of Chemical Physics•Apr 11, 2008•89 citations
We use the spin-1 Heisenberg chain with periodic boundary conditions to ilustrate that the systems get stuck in metastable configurations only when the density-matrix renormalization-group algorithm start with small number of states m. We also show that the convergence of the energies have a huge improvement if we start the algorithm with a large number of states m.
We develop a correction to the density matrix used in density matrix renormalization group calculations to take into account the incompleteness of the environment block. The correction allows successful calculations using only a single site in the center of the system, rather than the standard two sites, improving typical computation times by a factor of two to four. In addition, in many cases where the ordinary density matrix renormalization group method can get stuck in metastable configurations, the correction eliminates the sticking. We test the method on the Heisenberg S=1S=1S=1 chain.
A variant of White's density-matrix renormalization-group scheme, which is designed to compute low-lying energies of one-dimensional quantum lattice models with a large number of degrees of freedom per site is described. The method is tested on two exactly solvable models---the spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg chain and a dimerized XY\mathrm{XY}XY spin chain. To illustrate the potential of the method, it is applied to a model of spins interacting with quantum phonons. It is shown that the method accurately resolves a number of energy gaps on periodic rings that are sufficiently large to afford an accurate investigation of critical properties via the use of finite-size scaling theory.
The density matrix renormalization group (DMRG) method generates the low-energy states of linear systems of NNN sites with a few degrees of freedom at each site by starting with a small system and adding sites step by step while keeping constant the dimension of the truncated Hilbert space. DMRG algorithms are adapted to open chains with inversion symmetry at the central site, to cyclic chains and to weakly coupled chains. Physical properties rather than energy accuracy is the motivation. The algorithms are applied to the edge states of linear Heisenberg antiferromagnets with spin S≥1S \ge 1S≥1 and to the quantum phases of a frustrated spin-1/2 chain with exchange between first and second neighbors. The algorithms are found to be accurate for extended Hubbard and related 1D models with charge and spin degrees of freedom.
An improved trial-wave-function initialization approach is suggested in an infinite-chain density-matrix renormalization-group (DMRG) method. When we remove the sign freedom of the retained basis in a DMRG algorithm, an optimal trial wave function for a chain can be obtained from previous wave functions for chains with shorter lengths in DMRG. Such an optimal trial wave function will reduce the iteration number of Lanczos-style approaches in calculating the eigen-wave-functions. The possibilities of converging to other eigenenergy levels are also eliminated. A one-dimensional Heisenberg chain is used in this paper to illustrate our practice.
The density matrix renormalization group discovered by White is investigated. In the case where renormalization eventually converges to a fixed point we show that quantum states in the thermodynamic limit with periodic boundary conditions can be simply represented by a ``matrix product ground state'' with a natural description of Bloch states of elementary excitations. We then observe that these states can be rederived through a simple variational ansatz making no reference to a renormalization construction. The method is tested on the spin-1 Heisenberg model.
We present an approach that allows to produce the two-body density matrix during the density matrix renormalization group (DMRG) run without an additional increase in the current disk and memory requirements. The computational cost of producing the two-body density matrix is proportional to O(M3k2+M2k4). The method is based on the assumption that different elements of the two-body density matrix can be calculated during different steps of a sweep. Hence, it is desirable that the wave function at the convergence does not change during a sweep. We discuss the theoretical structure of the wave function ansatz used in DMRG, concluding that during the one-site DMRG procedure, the energy and the wave function are converging monotonically at every step of the sweep. Thus, the one-site algorithm provides an opportunity to obtain the two-body density matrix free from the N-representability problem. We explain the problem of local minima that may be encountered in the DMRG calculations. We discuss theoretically why and when the one- and two-site DMRG procedures may get stuck in a metastable solution, and we list practical solutions helping the minimization to avoid the local minima.