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

arXiv:cond-mat/9511083 · International Journal of Modern Physics A

ANALYTIC FORMULATIONS OF THE DENSITY MATRIX RENORMALIZATION GROUP

Miguel Angel Martin-Delgado, Germán Sierra

Published Jul 10, 1996• 67 citations
arXivHTMLPDF

Abstract

We present two new analytic formulations of the density matrix renormalization group (DMRG) method. In these formulations we combine the block renormalization group (BRG) procedure with the variational and Fokker-Planck methods. The BRG method is used to reduce the lattice size while the latter are used to construct approximate target states to compute the block density matrix. We apply our DMRG methods to the Ising model in a transverse field (ITF model) and compute several of its critical properties, which are then compared with the old BRG results.

DOI
10.1142/s0217751x96001516

Related papers

Accuracy of the density-matrix renormalization-group method

Örs Legeza, Gábor Fáth

White's density-matrix renormalization-group (DMRG) method has been applied to the one-dimensional Ising model in a transverse field (ITF), in order to study the accuracy of the numerical algorithm. Due to the exact solubility of the ITF for any finite chain length, the errors introduced by the basis truncation procedure could have been directly analyzed. By computing different properties, like the energies of the low-lying levels or the ground-state one- and two-point correlation functions, we obtained a detailed picture of how these errors behave as functions of the various model and algorithm parameters. Our experience with the ITF contributes to a better understanding of the DMRG method, and may facilitate its optimization in other applications.

arXiv:cond-mat/9809035•Physical review. B, Condensed matter•Jun 1, 1996•85 citations

Density Matrix Renormalization Group calculations for the Ising Model with a Transverse Field

Arnold H. Kole

In this thesis, the Density Matrix Renormalization Group algorithm is introduced. Some key concepts from quantum mechanics are discussed to aid in understanding the method and it is put into the context of the Renormalization Group. The algorithm is discussed from a mostly practical perspective. As a model to test the algorithm, the Ising Model with a Transverse Field was used. This model is interesting because it is one of the simplest one-dimensional models that displays a phase transition at a finite external field. It can also be solved analytically, which makes it possible to easily check the results of DMRG calculations. The DMRG algorithm was implemented from scratch in C++ for the Transverse Field Ising Model. Using this implementation, the errors in DMRG calculations were explored, as well as the phase transition in the Transverse Field Ising Chain.

•Jan 1, 2018•1 citation

The density matrix renormalization group and critical phenomena

Robert J. Bursill, Fatma Göde

We adapt White`s (1992, 1993) density matrix renormalization group (DMRG) to the direct study of critical phenomena. We use the DMRG to generate transformations in the space of coupling constants. We postulate that a study of density matrix eigenvalues leads to a natural identification of renormalized blocks, operators and Hamiltonians. We apply the scheme to the phase transition in the anisotropic spin-1/2 Heisenberg chain. In the simplest case where the two most probable states in odd-sized blocks are used to construct approximate renormalization group transformations, we find qualitative improvement upon the standard real-space renormalization group method for the thermal exponent nu .

Journal of Physics Condensed Matter•Dec 11, 1995•4 citations

Density-matrix renormalization using three classes of block states

Marie-Bernadette Lepetit, G. M. Pastor

An extension of the density-matrix renormalization-group (DMRG) method is presented. Besides the two groups or classes of block states considered in White's formulation, the retained m states, and the neglected ones, we introduce an intermediate group of block states having the following p largest eigenvalues ${\ensuremath{\lambda}}_{i}$ of the reduced density matrix: ${\ensuremath{\lambda}}_{1}>~\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}>~{\ensuremath{\lambda}}_{m}>~{\ensuremath{\lambda}}_{m+1}>~\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}>~{\ensuremath{\lambda}}_{m+p}.$ These states are taken into account when they contribute to intrablock transitions but are neglected when they participate in more delocalized interblock fluctuations. Applications to one-dimensional models (Heisenberg, Hubbard, and dimerized tight binding) show that in this way the involved computer resources can be reduced without significant loss of accuracy. The efficiency and accuracy of the method is analyzed by varying m and p and by comparison with standard DMRG calculations. A Hamiltonian-independent scheme for choosing m and p and for extrapolating to the limit where m and p are infinite is provided. Finally, an extension of the 3-classes approach is outlined, which incorporates the fluctuations between the p states of different blocks as a self-consistent dressing of the block interactions among the retained m states.

arXiv:cond-mat/9809233•Physical review. B, Condensed matter•Nov 15, 1998•3 citations

Equivalence of the variational matrix product method and the density matrix renormalization group applied to spin chains

J. Dukelsky, M. A Martín-Delgado, T Nishino +1 more

We study the relationship between the Density Matrix Renormalization Group (DMRG) and the variational matrix product method (MPM). In the latter method one can also define a density matrix whose eigenvalues turn out to be numerically close to those of the DMRG. We illustrate our ideas with the spin-1 Heisenberg chain, where we compute the ground-state energy and the spin correlation length. We also give a rotational invariant formulation of the MPM.

arXiv:cond-mat/9710310•Europhysics Letters (EPL)•Aug 15, 1998•173 citations

RECENT PROGRESS IN THE DENSITY-MATRIX RENORMALIZATION GROUP

Ulrich Schollwöck

Over the last decade, the density-matrix renormalization group (DMRG) has emerged as the most powerful method for the simulation of strongly correlated one-dimensional (1D) quantum systems. Input from quantum information has allowed to trace the method's performance to the entanglement properties of quantum states, revealing why it works so well in 1D and not so well in 2D; it has allowed to devise algorithms for time-dependent quantum systems and, by clarifying the link between DMRG and Wilson's numerical renormalization group (NRG), for quantum impurity systems.

International Journal of Modern Physics B•May 30, 2007•5 citations
We present two new analytic formulations of the density matrix renormalization group (DMRG) method. In these formulations we combine the block renormalization group (BRG) procedure with the variational and Fokker-Planck methods. The BRG method is used to reduce the lattice size while the latter are used to construct approximate target states to compute the block density matrix. We apply our DMRG methods to the Ising model in a transverse field (ITF model) and compute several of its critical properties, which are then compared with the old BRG results.
White's density-matrix renormalization-group (DMRG) method has been applied to the one-dimensional Ising model in a transverse field (ITF), in order to study the accuracy of the numerical algorithm. Due to the exact solubility of the ITF for any finite chain length, the errors introduced by the basis truncation procedure could have been directly analyzed. By computing different properties, like the energies of the low-lying levels or the ground-state one- and two-point correlation functions, we obtained a detailed picture of how these errors behave as functions of the various model and algorithm parameters. Our experience with the ITF contributes to a better understanding of the DMRG method, and may facilitate its optimization in other applications.
In this thesis, the Density Matrix Renormalization Group algorithm is introduced. Some key concepts from quantum mechanics are discussed to aid in understanding the method and it is put into the context of the Renormalization Group. The algorithm is discussed from a mostly practical perspective. As a model to test the algorithm, the Ising Model with a Transverse Field was used. This model is interesting because it is one of the simplest one-dimensional models that displays a phase transition at a finite external field. It can also be solved analytically, which makes it possible to easily check the results of DMRG calculations. The DMRG algorithm was implemented from scratch in C++ for the Transverse Field Ising Model. Using this implementation, the errors in DMRG calculations were explored, as well as the phase transition in the Transverse Field Ising Chain.
We adapt White`s (1992, 1993) density matrix renormalization group (DMRG) to the direct study of critical phenomena. We use the DMRG to generate transformations in the space of coupling constants. We postulate that a study of density matrix eigenvalues leads to a natural identification of renormalized blocks, operators and Hamiltonians. We apply the scheme to the phase transition in the anisotropic spin-1/2 Heisenberg chain. In the simplest case where the two most probable states in odd-sized blocks are used to construct approximate renormalization group transformations, we find qualitative improvement upon the standard real-space renormalization group method for the thermal exponent nu .
An extension of the density-matrix renormalization-group (DMRG) method is presented. Besides the two groups or classes of block states considered in White's formulation, the retained m states, and the neglected ones, we introduce an intermediate group of block states having the following p largest eigenvalues ${\ensuremath{\lambda}}_{i}$ of the reduced density matrix: ${\ensuremath{\lambda}}_{1}>~\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}>~{\ensuremath{\lambda}}_{m}>~{\ensuremath{\lambda}}_{m+1}>~\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}>~{\ensuremath{\lambda}}_{m+p}.$ These states are taken into account when they contribute to intrablock transitions but are neglected when they participate in more delocalized interblock fluctuations. Applications to one-dimensional models (Heisenberg, Hubbard, and dimerized tight binding) show that in this way the involved computer resources can be reduced without significant loss of accuracy. The efficiency and accuracy of the method is analyzed by varying m and p and by comparison with standard DMRG calculations. A Hamiltonian-independent scheme for choosing m and p and for extrapolating to the limit where m and p are infinite is provided. Finally, an extension of the 3-classes approach is outlined, which incorporates the fluctuations between the p states of different blocks as a self-consistent dressing of the block interactions among the retained m states.
Over the last decade, the density-matrix renormalization group (DMRG) has emerged as the most powerful method for the simulation of strongly correlated one-dimensional (1D) quantum systems. Input from quantum information has allowed to trace the method's performance to the entanglement properties of quantum states, revealing why it works so well in 1D and not so well in 2D; it has allowed to devise algorithms for time-dependent quantum systems and, by clarifying the link between DMRG and Wilson's numerical renormalization group (NRG), for quantum impurity systems.