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/9706124 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Density-matrix renormalization group for the Berezinskii-Kosterlitz-Thouless transition of the 19-vertex model

Yasushi Honda, Tsuyoshi Horiguchi

Published Oct 1, 1997• 19 citations
arXivHTMLPDF

Abstract

We embody the density-matrix renormalization-group (DMRG) method for the 19-vertex model on a square lattice in order to investigate the Berezinskii-Kosterlitz-Thouless transition. Elements of the transfer matrix of the 19-vertex model are classified in terms of the total value of arrows in one layer of the square lattice. By using this classification, we succeed in reducing enormously the dimension of the matrix that has to be diagonalized in the DMRG method. We apply our method to the 19-vertex model with the interaction $K=1.0866$ and obtain $c=1.006(1)$ for the conformal anomaly.

DOI
10.1103/physreve.56.3920

Related papers

Critical behavior of a 19-vertex model with full frustration

Yasushi Honda, Tsuyoshi Horiguchi

We apply the singular value decomposition method to matrices made from eigenvectors of the transfer matrix of a 19-vertex model on a square lattice with full frustration. The conformal anomaly c is estimated to be close to 3/2. This result suggests that the 19-vertex model with full frustration has a single transition due to two kinds of degrees of freedom, each of which has U(1) symmetry and ${Z}_{2}$ symmetry. Moreover, we find that the value of index $\ensuremath{\eta}$ is smaller than that at the Berezinskii-Kosterlitz-Thouless (BKT) transition by considering a logarithmic correction to the finite size analysis. This result for $\ensuremath{\eta}$ suggests that there is another universality class which is different from that of either the Onsager type or the BKT type.

Physical review. B, Condensed matter•Nov 1, 1998•7 citations

Density Matrix Renormalization Group Method for 2D Classical Models

Tomotoshi Nishino

The density matrix renormalization group (DMRG) method is applied to the interaction round a face (IRF) model. When the transfer matrix is asymmetric, singular-value decomposition of the density matrix is required. A trial numerical calculation is performed on the square lattice Ising model, which is a special case of the IRF model.

arXiv:cond-mat/9508111•Journal of the Physical Society of Japan•Oct 1, 1995•267 citations

Two-dimensional algorithm of the density-matrix renormalization group

Tao Xiang, Jizhong Lou, Zhao-Bin Su

We propose an approach to implement the density-matrix renormalization group (DMRG) in two dimensions. With this approach the initial blocks of a $L\ifmmode\times\else\texttimes\fi{}L$ lattice are built up directly from the matrix elements of a $(L\ensuremath{-}1)\ifmmode\times\else\texttimes\fi{}(L\ensuremath{-}1)$ lattice and the topological characteristics of two-dimensional lattices are preserved in the iteration of DMRG. By applying it to the spin-1/2 Heisenberg model on both square and triangular lattices, we find that this approach is significantly more efficient and accurate than other two-dimensional DMRG methods currently in use.

arXiv:cond-mat/0102200•Physical review. B, Condensed matter•Aug 21, 2001•76 citations

96 The Density Matrix Renormalization Group Method applied to Interaction Round a Face Hamiltonians

Tomotoshi Nishino

Given a Hamiltonian with a continuous symmetry one can generally factorize that symmetry and consider the dynamics on invariant Hilbert Spaces. In Statistical Mechanics this procedure is known as the vertex-IRF map, and in certain cases, like rotational invariant Hamiltonians, can be implemented via group theoretical techniques. Using this map we translate the DMRG method, which applies to 1d vertex Hamiltonians, into a formulation adequate to study IRF Hamiltonians. The advantage of the IRF formulation of the DMRG method ( we name it IRF-DMRG), is that the dimensions of the Hilbert Spaces involved in numerical computations are smaller than in the vertex-DMRG, since the degeneracy due to the symmetry has been eliminated. The IRF-DMRG admits a natural and geometric formulation in terms of the paths or string algebras used in Exactly Integrable Systems and Conformal Field Theory. We illustrate the IRF-DMRG method with the study of the SOS model which corresponds to the spin 1/2 Heisenberg chain and the RSOS models with Coxeter diagram of type A, which correspond to the quantum group invariant XXZ chain.

•Jan 1, 1996•31 citations

Renormalization group method for quasi-one-dimensional quantum Hamiltonians

S. Moukouri, Laurent G. Caron

A density-matrix renormalization-group (DMRG) method for highly anisotropic two-dimensional (2D) systems is presented. The method consists of applying the usual DMRG in two steps. In the first step, a pure one-dimensional calculation along the longitudinal direction is made in order to generate a low-energy Hamiltonian. In the second step, the anisotropic 2D lattice is obtained by coupling in the transverse direction the 1D Hamiltonians. The method is applied to the anisotropic quantum spin-half Heisenberg model on a square lattice.

arXiv:cond-mat/0210668•Physical review. B, Condensed matter•Mar 27, 2003•15 citations

Accurate localization of Kosterlitz-Thouless-type quantum phase transitions for one-dimensional spinless fermions

Florian Gebhard, Kevin Bauerbach, Örs Legeza

Spinless fermions in one dimension with nearest-neighbor transfer amplitude $\ensuremath{-}t$ and interaction $V$ display a Kosterlitz-Thouless-type quantum phase transition from a Luttinger liquid metal to a charge density wave insulator at finite interaction strength ${V}_{\mathrm{c}}=2t$. The authors use the density matrix renormalization group (DMRG) method on rings with $L\ensuremath{\le}512$ sites for an accurate localization of the transition. The maximum of the quasiparticle density in the upper Hartree-Fock band, ${n}_{\ensuremath{\beta}}^{\mathrm{max}}(L,V)={\mathrm{max}}_{k}\phantom{\rule{0.167em}{0ex}}{n}_{k,\ensuremath{\beta}}(L,V)$, develops a peak close to the critical interaction whose position extrapolates to the exact result in the thermodynamic limit, ${lim}_{L\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}{V}_{\ensuremath{\beta}}^{\mathrm{max}}(L,V)={V}_{\mathrm{c}}$, with an accuracy of better than one percent. Monitoring the Luttinger parameter permits to locate the transition with an accuracy of three percent.

arXiv:2208.07620•Physical review. B./Physical review. B•Nov 18, 2022•8 citations
We embody the density-matrix renormalization-group (DMRG) method for the 19-vertex model on a square lattice in order to investigate the Berezinskii-Kosterlitz-Thouless transition. Elements of the transfer matrix of the 19-vertex model are classified in terms of the total value of arrows in one layer of the square lattice. By using this classification, we succeed in reducing enormously the dimension of the matrix that has to be diagonalized in the DMRG method. We apply our method to the 19-vertex model with the interaction K=1.0866K=1.0866K=1.0866 and obtain c=1.006(1)c=1.006(1)c=1.006(1) for the conformal anomaly.
We apply the singular value decomposition method to matrices made from eigenvectors of the transfer matrix of a 19-vertex model on a square lattice with full frustration. The conformal anomaly c is estimated to be close to 3/2. This result suggests that the 19-vertex model with full frustration has a single transition due to two kinds of degrees of freedom, each of which has U(1) symmetry and Z2{Z}_{2}Z2​ symmetry. Moreover, we find that the value of index $\ensuremath{\eta}$ is smaller than that at the Berezinskii-Kosterlitz-Thouless (BKT) transition by considering a logarithmic correction to the finite size analysis. This result for $\ensuremath{\eta}$ suggests that there is another universality class which is different from that of either the Onsager type or the BKT type.
We propose an approach to implement the density-matrix renormalization group (DMRG) in two dimensions. With this approach the initial blocks of a $L\ifmmode\times\else\texttimes\fi{}L$ lattice are built up directly from the matrix elements of a $(L\ensuremath{-}1)\ifmmode\times\else\texttimes\fi{}(L\ensuremath{-}1)$ lattice and the topological characteristics of two-dimensional lattices are preserved in the iteration of DMRG. By applying it to the spin-1/2 Heisenberg model on both square and triangular lattices, we find that this approach is significantly more efficient and accurate than other two-dimensional DMRG methods currently in use.
Given a Hamiltonian with a continuous symmetry one can generally factorize that symmetry and consider the dynamics on invariant Hilbert Spaces. In Statistical Mechanics this procedure is known as the vertex-IRF map, and in certain cases, like rotational invariant Hamiltonians, can be implemented via group theoretical techniques. Using this map we translate the DMRG method, which applies to 1d vertex Hamiltonians, into a formulation adequate to study IRF Hamiltonians. The advantage of the IRF formulation of the DMRG method ( we name it IRF-DMRG), is that the dimensions of the Hilbert Spaces involved in numerical computations are smaller than in the vertex-DMRG, since the degeneracy due to the symmetry has been eliminated. The IRF-DMRG admits a natural and geometric formulation in terms of the paths or string algebras used in Exactly Integrable Systems and Conformal Field Theory. We illustrate the IRF-DMRG method with the study of the SOS model which corresponds to the spin 1/2 Heisenberg chain and the RSOS models with Coxeter diagram of type A, which correspond to the quantum group invariant XXZ chain.
A density-matrix renormalization-group (DMRG) method for highly anisotropic two-dimensional (2D) systems is presented. The method consists of applying the usual DMRG in two steps. In the first step, a pure one-dimensional calculation along the longitudinal direction is made in order to generate a low-energy Hamiltonian. In the second step, the anisotropic 2D lattice is obtained by coupling in the transverse direction the 1D Hamiltonians. The method is applied to the anisotropic quantum spin-half Heisenberg model on a square lattice.
Spinless fermions in one dimension with nearest-neighbor transfer amplitude $\ensuremath{-}t$ and interaction VVV display a Kosterlitz-Thouless-type quantum phase transition from a Luttinger liquid metal to a charge density wave insulator at finite interaction strength Vc=2t{V}_{\mathrm{c}}=2tVc​=2t. The authors use the density matrix renormalization group (DMRG) method on rings with $L\ensuremath{\le}512$ sites for an accurate localization of the transition. The maximum of the quasiparticle density in the upper Hartree-Fock band, ${n}_{\ensuremath{\beta}}^{\mathrm{max}}(L,V)={\mathrm{max}}_{k}\phantom{\rule{0.167em}{0ex}}{n}_{k,\ensuremath{\beta}}(L,V)$, develops a peak close to the critical interaction whose position extrapolates to the exact result in the thermodynamic limit, ${lim}_{L\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}{V}_{\ensuremath{\beta}}^{\mathrm{max}}(L,V)={V}_{\mathrm{c}}$, with an accuracy of better than one percent. Monitoring the Luttinger parameter permits to locate the transition with an accuracy of three percent.