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/0106194 · Journal of Physics A Mathematical and General

On the choice of the density matrix in the stochastic TMRG

T Enss, U Schollwöck

Published Sep 14, 2001• 31 citations
arXivHTMLPDF

Abstract

In applications of the density matrix renormalization group to nonhermitean problems, the choice of the density matrix is not uniquely prescribed by the algorithm. We demonstrate that for the recently introduced stochastic transfer matrix DMRG (stochastic TMRG) the necessity to use open boundary conditions makes asymmetrical reduced density matrices, as used for renormalization in quantum TMRG, an inappropriate choice. An explicit construction of the largest left and right eigenvectors of the full transfer matrix allows us to show why symmetrical density matrices are the correct physical choice.

DOI
10.1088/0305-4470/34/38/305

Related papers

The density-matrix renormalization group

Ulrich Schollwöck

The density-matrix renormalization group (DMRG) is a numerical algorithm for the efficient truncation of the Hilbert space of low-dimensional strongly correlated quantum systems based on a rather general decimation prescription. This algorithm has achieved unprecedented precision in the description of one-dimensional quantum systems. It has therefore quickly become the method of choice for numerical studies of such systems. Its applications to the calculation of static, dynamic, and thermodynamic quantities in these systems are reviewed here. The potential of DMRG applications in the fields of two-dimensional quantum systems, quantum chemistry, three-dimensional small grains, nuclear physics, equilibrium and nonequilibrium statistical physics, and time-dependent phenomena is also discussed. This review additionally considers the theoretical foundations of the method, examining its relationship to matrix-product states and the quantum information content of the density matrices generated by the DMRG.

arXiv:cond-mat/0409292•Reviews of Modern Physics•Apr 26, 2005•3,323 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

Efficiency of Asymmetric Targeting for Finite-TDMRG

Tomotoshi Nishino, Naokazu Shibata

Two targeting schemes have been known for the density matrix renormalization group (DMRG) applied to non-Hermitian problems; one uses an asymmetric density matrix and the other uses symmetric density matrix. We compare the numerical efficiency of these two targeting schemes when they are used for the finite temperature DMRG.

arXiv:cond-mat/9905229•Journal of the Physical Society of Japan•Nov 15, 1999•8 citations

Transfer-matrix density-matrix renormalization-group theory for thermodynamics of one-dimensional quantum systems

Xiaoqun Wang, Tao Xiang

The transfer-matrix density-matrix renormalization-group 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 ${10}^{\ensuremath{-}3}$ down to $T=0.01J$ with 80 states kept.

Physical review. B, Condensed matter•Sep 1, 1997•273 citations

Biorthonormal transfer-matrix renormalization-group method for non-Hermitian matrices

Yukun Huang

A biorthonormal transfer-matrix renormalization-group (BTMRG) method for non-Hermitian matrices is presented. This BTMRG produces a dual set of biorthonormal bases to construct the renormalized transfer matrix with only half the dimensions of the matrix of a conventional transfer-matrix renormalization group (TMRG). We show that under generic conditions, such biorthonormal bases always exist. Based on a special E·S·E scheme (where S and E represent the system and environment blocks, respectively, and the two dots in between represent two additional physical sites), the BTMRG method can achieve zero truncation of any reduced state in describing both current left and right Perron states so as to reach a high degree of efficiency and accuracy. We believe that the BTMRG constitutes a more powerful and robust tool than conventional TMRG for non-Hermitian matrices and that it would allow us to better understand the collective behaviors and emerging phenomena of strongly correlated many-body systems. We also show that this scheme is particularly adapted to the calculation of the two-site correlation function of a one-dimensional quantum or two-dimensional classical lattice model.

Physical Review E•Mar 10, 2011•13 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
In applications of the density matrix renormalization group to nonhermitean problems, the choice of the density matrix is not uniquely prescribed by the algorithm. We demonstrate that for the recently introduced stochastic transfer matrix DMRG (stochastic TMRG) the necessity to use open boundary conditions makes asymmetrical reduced density matrices, as used for renormalization in quantum TMRG, an inappropriate choice. An explicit construction of the largest left and right eigenvectors of the full transfer matrix allows us to show why symmetrical density matrices are the correct physical choice.
The density-matrix renormalization group (DMRG) is a numerical algorithm for the efficient truncation of the Hilbert space of low-dimensional strongly correlated quantum systems based on a rather general decimation prescription. This algorithm has achieved unprecedented precision in the description of one-dimensional quantum systems. It has therefore quickly become the method of choice for numerical studies of such systems. Its applications to the calculation of static, dynamic, and thermodynamic quantities in these systems are reviewed here. The potential of DMRG applications in the fields of two-dimensional quantum systems, quantum chemistry, three-dimensional small grains, nuclear physics, equilibrium and nonequilibrium statistical physics, and time-dependent phenomena is also discussed. This review additionally considers the theoretical foundations of the method, examining its relationship to matrix-product states and the quantum information content of the density matrices generated by the DMRG.
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 transfer-matrix density-matrix renormalization-group 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 ${10}^{\ensuremath{-}3}$ down to T=0.01JT=0.01JT=0.01J with 80 states kept.
A biorthonormal transfer-matrix renormalization-group (BTMRG) method for non-Hermitian matrices is presented. This BTMRG produces a dual set of biorthonormal bases to construct the renormalized transfer matrix with only half the dimensions of the matrix of a conventional transfer-matrix renormalization group (TMRG). We show that under generic conditions, such biorthonormal bases always exist. Based on a special E·S·E scheme (where S and E represent the system and environment blocks, respectively, and the two dots in between represent two additional physical sites), the BTMRG method can achieve zero truncation of any reduced state in describing both current left and right Perron states so as to reach a high degree of efficiency and accuracy. We believe that the BTMRG constitutes a more powerful and robust tool than conventional TMRG for non-Hermitian matrices and that it would allow us to better understand the collective behaviors and emerging phenomena of strongly correlated many-body systems. We also show that this scheme is particularly adapted to the calculation of the two-site correlation function of a one-dimensional quantum or two-dimensional classical lattice model.
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.