Application of the density matrix renormalization group method to finite temperatures and two-dimensional systems

Shibata, Naokazu

doi:10.1088/0305-4470/36/37/201

Cited by 44 publications

(50 citation statements)

References 65 publications

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“…As the dimension of QTM exponentially increases with increasing Trotter number computations of Z M are feasible for relatively small M which can prevent reliable estimations of the thermodynamic functions in a low-temperature region (see references in [12][13][14]). To overcome this restriction and cover the entire experimental temperature range, the DMRG approach has been applied [15][16][17][18].…”

Section: Simulation Methodsmentioning

confidence: 99%

“…In the finite-temperature DMRG method, we extend the transfer matrix in the Trotter direction by restricting the basis states using the density matrix calculated from the transfer matrix eigenvector related to the maximum eigenvalue λ max [18]. Contrary to the zerotemperature DMRG method, the transfer matrices to be diagonalized are asymmetric here due to the checkerboard decomposition.…”

Section: Simulation Methodsmentioning

confidence: 99%

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Quantum effects and Haldane gap in magnetic chains with alternating anisotropy axes

Barasiński

Drzewiński

Kamieniarz

2011

Computer Physics Communications

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Section: Simulation Methodsmentioning

confidence: 99%

Section: Simulation Methodsmentioning

confidence: 99%

Quantum effects and Haldane gap in magnetic chains with alternating anisotropy axes

Barasiński

Drzewiński

Kamieniarz

2011

Computer Physics Communications

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“…where S is a classical Coulomb energy and c † n creates an electron in the single-particle state n = 1, ..., M, M being the total number of single-particle states in the unit cell, and A is the interaction matrix of the Coulomb force (for details see [32]). …”

Section: D Electrons In a High Magnetic Fieldmentioning

confidence: 99%

“…Several other important applications of the extended methodology have also been reported recently, including applications to electrons in the lowest three Landau levels [29,30,31,32,33]. The DMRG methodology, both in real space and in the versions appropriate to finite Fermi systems, usually violates some symmetries of the underlying Hamiltonian.…”

Section: Introductionmentioning

confidence: 99%

The density matrix renormalization group for finite fermi systems

Dukelsky

P̧ittel

2004

Rep. Prog. Phys.

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Abstract. The Density Matrix Renormalization Group (DMRG) was introduced by Steven White in 1992 as a method for accurately describing the properties of onedimensional 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 realspace 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.

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“…3 and 4). It is also possible to calculate dynamic properties [3][4][5][6][7] and work at finite-temperature through the DMRG [8][9][10]. The main advantage of DMRG, compared with the Lanczos exact diagonalization [11], is its capability to obtain the ground-state properties of very large systems in a well controlled way.…”

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confidence: 99%

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

Xavier

2009

Braz. J. Phys.

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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. Keywords: spin model, DMRG, The density-matrix renormalization-group [1, 2] (DMRG) is one of the most appropriate techniques to study static properties of the one-dimensional systems at zero temperature (for a review see, for example, Refs. 3 and 4). It is also possible to calculate dynamic properties [3][4][5][6][7] and work at finite-temperature through the DMRG [8][9][10]. The main advantage of DMRG, compared with the Lanczos exact diagonalization [11], is its capability to obtain the ground-state properties of very large systems in a well controlled way. Note that it is also possible to investigate large systems by Monte Carlo methods. However, the Monte Carlo technique is not appropriated to study frustrated/fermionic systems due to the "sign" problem.Although the DMRG algorithm was developed for onedimensional systems, it has been used to treat twodimensional systems [12][13][14][15][16][17]. The procedure consists in mapping the low-dimensional model on an one-dimensional model with long range interactions. As first point out by Liang and Pang [12], the energies of two-dimensional systems, converge slower than the ones of one-dimensional systems with short range interactions. The number of states needed to keep a fixed accuracy seems to increase exponentially with the width of the system. A similar effect also appears when we study onedimensional systems with periodic boundary condition (PBC) [1,2]. Since the DMRG was developed, it was observed that the ground state energy (as a function of the number of states m kept in the truncation process) converge faster for the system with open boundary condition (OBC) than the one with PBC. Although it is not completely understood, it seems that everytime that an operator that acts in the left block is directly connected with an operator that acts in the right block (see Fig. 1), the ground state energy convergence is slower. This has been observed for one-dimensional systems as well for the two-dimensional systems.Another difficulty also appears when the left and right blocks are directly connected. Some times, in the simulations, the system gets stuck in some local minimum of energy (see Fig. 3(a)), even working with large values of m [18,19]. This is a serious problem. If the energies do not change increasing m, we may think naively that the true ground state energy was reached. But in fact, the energy found is far from the true ground state. decreasing the computation time as well as the memory used, and more important, (ii) to avoid that the system gets stuck in metastable configurations. As discussed by White [19], the main reason that the simulations get stuck in metastable configurations is due...

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Application of the density matrix renormalization group method to finite temperatures and two-dimensional systems

Cited by 44 publications

References 65 publications

Quantum effects and Haldane gap in magnetic chains with alternating anisotropy axes

Quantum effects and Haldane gap in magnetic chains with alternating anisotropy axes

The density matrix renormalization group for finite fermi systems

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

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