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

QUANTUM CORNER — TRANSFER MATRIX DMRG

Erik Bartel, Andreas Schadschneider

Published Jan 1, 2008• 2 citations
OpenAlex

Abstract

We propose a new method for the calculation of thermodynamic properties of one-dimensional quantum systems by combining the TMRG approach with the corner transfer-matrix method. The corner transfer-matrix DMRG method brings reasonable advantage over TMRG for classical systems. We have modified the concept for the calculation of thermal properties of one-dimensional quantum systems. The novel QCTMRG algorithm is implemented and used to study two simple test cases, the classical Ising chain and the isotropic Heisenberg model. In a discussion, the advantages and challenges are illuminated.

Related papers

QUANTUM CORNER — TRANSFER MATRIX DMRG

ERIK BARTEL, ANDREAS SCHADSCHNEIDER

We propose a new method for the calculation of thermodynamic properties of one-dimensional quantum systems by combining the TMRG approach with the corner transfer-matrix method. The corner transfer-matrix DMRG method brings reasonable advantage over TMRG for classical systems. We have modified the concept for the calculation of thermal properties of one-dimensional quantum systems. The novel QCTMRG algorithm is implemented and used to study two simple test cases, the classical Ising chain and the isotropic Heisenberg model. In a discussion, the advantages and challenges are illuminated.

arXiv:0804.1874•International Journal of Modern Physics C•Aug 1, 2008•3 citations

The density-matrix renormalization group applied to transfer matrices: Static and dynamical properties of one-dimensional quantum systems at finite temperature

S. Glocke, Andreas Klümper, Jesko Sirker

The density-matrix renormalization group (DMRG) applied to transfer matrices allows it to calculate static as well as dynamical properties of one-dimensional quantum systems at finite temperature in the thermodynamic limit. To this end the quantum system is mapped onto a two-dimensional classical system by a Trotter-Suzuki decomposition. Here we discuss two different mappings: The standard mapping onto a two-dimensional lattice with checkerboard structure as well as an alternative mapping introduced by two of us. For the classical system an appropriate quantum transfer matrix is defined which is then treated using a DMRG scheme. As applications, the calculation of thermodynamic properties for a spin-1/2 Heisenberg chain in a staggered magnetic field and the calculation of boundary contributions for open spin chains are discussed. Finally, we show how to obtain real time dynamics from a classical system with complex Boltzmann weights and present results for the autocorrelation function of the XXZ-chain.

arXiv:cond-mat/0610689•Oct 25, 2006•0 citations

Transfermatrix-DMRG for dynamics of stochastic models and thermodynamics of fermionic models

Andreas Kemper

The present work applies a numerical method, namely the transfer-matrix density-matrix renormalization group (TMRG), to two seemingly different types of models. In a first part the TMRG is used to investigate the thermodynamics of one-dimensional fermionic models. A second part deals with a novel TMRG method for one-dimensional stochastic models, whose development is an integral part of the thesis. First, the traditional TMRG algorithm for quantum systems is outlined in its historical context. Two different variants are presented, following works of Xiang et al. and Sirker and Klümper, respectively. The basic idea of the method is to map the thermodynamics of a one-dimensional quantum model by Trotter-Suzuki decomposition onto a two-dimensional statistical one. The latter is then solved by a transfer-matrix approach combined with the iterative numerical procedure of White's density-matrix renormalization-group (DMRG) algorithm. Thereby precise computations of various thermodynamic properties, such as thermodynamic potentials, susceptibilities, thermal expectation values and correlation functions are possible. As the first part of the thesis deals with fermionic models, we next review some basics about the theory of strongly correlated fermions in one dimension. Thereupon we elucidate the so-called Hirsch model, which recently gained a lot of theoretic interest in respect to high-temperature superconductivity. It extends the well-studied Hubbard model by an off-diagonal bond-charge interaction term. The current state of research is briefly summarized and mainly refers to ground state properties. Showing numerical TMRG results we then investigate and discuss the almost unknown thermodynamics of the Hirsch model. Various phases are identified and characterized in terms of Tomonaga-Luttinger and Luther-Emery liquid properties, in accordance with previous studies of the ground state. As an important result, superconducting singlet-pair correlation lenths are observed to dominate the physics at finite temperatures in a certain spin-gaped phase. Subsequent to our thermodynamic studies, we turn to the second part of the thesis and outline some theoretic basics of stochastic models. Most notably is the important formal analogy of the master equation, that describes the dynamics of the model, to a Schrödinger equation in imaginary time. This analogy is used to construct a stochastic TMRG algorithm almost similar to the quantum case, that facilitates the computation of dynamic properties, e.g. the local density of particles. We intensively focus on interesting mathematical properties of the stochastic transfer-matrix. As an astonishing result it is found, that the temporal evolution of the non-equilibrium process is reflected by a certain causal structure of the stochastic TMRG. But even if this new approach seems to be promising at first glance, severe numerical problems limit significantly its practical use. In order to solve these instabilities we propose a completely new variant of the algorithm, which we call stochastic light-cone corner-transfermatrix DMRG (LCTMRG). As suggested by its name, the LCTMRG makes use of the causal structure mentioned above and combines it with the stochastic TMRG algorithm. Applications of the LCTMRG onto various reaction-diffusion models verify highly precise numerical data and a great improve compared to the old algorithm by several orders of magnitude. Additionally it is stressed, that the newly proposed analysis tool provides some considerable advantages to common simulation techniques.

•Jan 1, 2003•1 citation

Classical Transfer Matrix Renormalization

Tao Xiang

This chapter introduces two kinds of RG methods for solving the leading eigenvalue and eigenvectors of a transfer matrix: TMRG (transfer matrix renormalization group) and CTMRG (corner transfer matrix renormalization group). These methods are developed to study the thermodynamic properties of two-dimensional classical statistical models. Furthermore, in the framework of MPS, the fixed-point equations of these methods are derived, and the steps for efficiently solving these equations are outlined.

Cambridge University Press eBooks•Aug 31, 2023•0 citations

Transfer Matrix DMRG for Thermodynamics of One-Dimensional Quantum Systems

Xiaoqun Wang, Tao Xiang

The transfer matrix DMRG method for one dimensional quantum lattice systems has been developed by considering the symmetry property of the transfer matrix and introducing the asymmetric reduced density matrix. We have evaluated a number of thermodynamic quantities of the anisotropic spin-1/2 Heisenberg model using this method and found that the results agree very accurately with the exact ones. The relative errors for the spin susceptibility are less than $10^{-3}$ down to $T=0.01J$ with 80 states kept.

arXiv:cond-mat/9705301•May 29, 1997•5 citations

Density-matrix spectra for integrable models

Ingo Peschel, Matthias Kaulke, Örs Legeza

The spectra which occur in numerical density-matrix renormalization group (DMRG) calculations for quantum chains can be obtained analytically for integrable models via corner transfer matrices. This is shown in detail for the transverse Ising chain and the uniaxial XXZ Heisenberg model and explains in particular their exponential character in these cases.

arXiv:cond-mat/9810174•Annalen der Physik•Feb 1, 1999•151 citations
We propose a new method for the calculation of thermodynamic properties of one-dimensional quantum systems by combining the TMRG approach with the corner transfer-matrix method. The corner transfer-matrix DMRG method brings reasonable advantage over TMRG for classical systems. We have modified the concept for the calculation of thermal properties of one-dimensional quantum systems. The novel QCTMRG algorithm is implemented and used to study two simple test cases, the classical Ising chain and the isotropic Heisenberg model. In a discussion, the advantages and challenges are illuminated.
We propose a new method for the calculation of thermodynamic properties of one-dimensional quantum systems by combining the TMRG approach with the corner transfer-matrix method. The corner transfer-matrix DMRG method brings reasonable advantage over TMRG for classical systems. We have modified the concept for the calculation of thermal properties of one-dimensional quantum systems. The novel QCTMRG algorithm is implemented and used to study two simple test cases, the classical Ising chain and the isotropic Heisenberg model. In a discussion, the advantages and challenges are illuminated.
The density-matrix renormalization group (DMRG) applied to transfer matrices allows it to calculate static as well as dynamical properties of one-dimensional quantum systems at finite temperature in the thermodynamic limit. To this end the quantum system is mapped onto a two-dimensional classical system by a Trotter-Suzuki decomposition. Here we discuss two different mappings: The standard mapping onto a two-dimensional lattice with checkerboard structure as well as an alternative mapping introduced by two of us. For the classical system an appropriate quantum transfer matrix is defined which is then treated using a DMRG scheme. As applications, the calculation of thermodynamic properties for a spin-1/2 Heisenberg chain in a staggered magnetic field and the calculation of boundary contributions for open spin chains are discussed. Finally, we show how to obtain real time dynamics from a classical system with complex Boltzmann weights and present results for the autocorrelation function of the XXZ-chain.
The present work applies a numerical method, namely the transfer-matrix density-matrix renormalization group (TMRG), to two seemingly different types of models. In a first part the TMRG is used to investigate the thermodynamics of one-dimensional fermionic models. A second part deals with a novel TMRG method for one-dimensional stochastic models, whose development is an integral part of the thesis. First, the traditional TMRG algorithm for quantum systems is outlined in its historical context. Two different variants are presented, following works of Xiang et al. and Sirker and Klümper, respectively. The basic idea of the method is to map the thermodynamics of a one-dimensional quantum model by Trotter-Suzuki decomposition onto a two-dimensional statistical one. The latter is then solved by a transfer-matrix approach combined with the iterative numerical procedure of White's density-matrix renormalization-group (DMRG) algorithm. Thereby precise computations of various thermodynamic properties, such as thermodynamic potentials, susceptibilities, thermal expectation values and correlation functions are possible. As the first part of the thesis deals with fermionic models, we next review some basics about the theory of strongly correlated fermions in one dimension. Thereupon we elucidate the so-called Hirsch model, which recently gained a lot of theoretic interest in respect to high-temperature superconductivity. It extends the well-studied Hubbard model by an off-diagonal bond-charge interaction term. The current state of research is briefly summarized and mainly refers to ground state properties. Showing numerical TMRG results we then investigate and discuss the almost unknown thermodynamics of the Hirsch model. Various phases are identified and characterized in terms of Tomonaga-Luttinger and Luther-Emery liquid properties, in accordance with previous studies of the ground state. As an important result, superconducting singlet-pair correlation lenths are observed to dominate the physics at finite temperatures in a certain spin-gaped phase. Subsequent to our thermodynamic studies, we turn to the second part of the thesis and outline some theoretic basics of stochastic models. Most notably is the important formal analogy of the master equation, that describes the dynamics of the model, to a Schrödinger equation in imaginary time. This analogy is used to construct a stochastic TMRG algorithm almost similar to the quantum case, that facilitates the computation of dynamic properties, e.g. the local density of particles. We intensively focus on interesting mathematical properties of the stochastic transfer-matrix. As an astonishing result it is found, that the temporal evolution of the non-equilibrium process is reflected by a certain causal structure of the stochastic TMRG. But even if this new approach seems to be promising at first glance, severe numerical problems limit significantly its practical use. In order to solve these instabilities we propose a completely new variant of the algorithm, which we call stochastic light-cone corner-transfermatrix DMRG (LCTMRG). As suggested by its name, the LCTMRG makes use of the causal structure mentioned above and combines it with the stochastic TMRG algorithm. Applications of the LCTMRG onto various reaction-diffusion models verify highly precise numerical data and a great improve compared to the old algorithm by several orders of magnitude. Additionally it is stressed, that the newly proposed analysis tool provides some considerable advantages to common simulation techniques.
This chapter introduces two kinds of RG methods for solving the leading eigenvalue and eigenvectors of a transfer matrix: TMRG (transfer matrix renormalization group) and CTMRG (corner transfer matrix renormalization group). These methods are developed to study the thermodynamic properties of two-dimensional classical statistical models. Furthermore, in the framework of MPS, the fixed-point equations of these methods are derived, and the steps for efficiently solving these equations are outlined.
The transfer matrix DMRG method for one dimensional quantum lattice systems has been developed by considering the symmetry property of the transfer matrix and introducing the asymmetric reduced density matrix. We have evaluated a number of thermodynamic quantities of the anisotropic spin-1/2 Heisenberg model using this method and found that the results agree very accurately with the exact ones. The relative errors for the spin susceptibility are less than 10−310^{-3}10−3 down to T=0.01JT=0.01JT=0.01J with 80 states kept.