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arXiv:cond-mat/9809035 · Physical review. B, Condensed matter

Accuracy of the density-matrix renormalization-group method

Örs Legeza, Gábor Fáth

Published Jun 1, 1996• 85 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.53.14349

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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.
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.
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.
The density matrix renormalization group (DMRG) is a numerical method for studying low dimensional strongly correlated models. The study of these models is one of the most active areas of condensed matter physics today and has been spurred not only by the discovery of high temperature superconductivity but also by experiments on new compounds. In this thesis we study fundamentals of the density matrix renormalization group method as well as applications of the method to Heisenberg spin chains. The introductory text is meant as a brief overview of the DMRG as well as Heisenberg spin systems and provides an introduction to the papers. In papers I and II we investigate the density matrix renormalization group with the aim to better understand the underlying structure of the method. We find that if the renormalization converges to a fixed point in the thermodynamic limit, the resulting basis states can be written as a matrix product form. We also show an explicit relation of the DMRG to variational calculations using the fact that the matrix product states can be rederived through a variational ansatz without any reference to the DMRG calculation. It is then shown how the matrix product states can be used to define a set of Bloch waves for describing elementary excitations of the Heisenberg spin-1 chain. The low lying energy spectrum is calculated. In papers III and IV we investigate the thermodynamics of impurities in a spin-1/2 chain. For this purpose we have shown that a recently developed transfer matrix DMRG method can be used to calculate impurity properties. Both impurity corrections to thermodynamic properties and local properties at finite temperature can be obtained. The method gives quicker and more accurate results than quantum Monte Carlo. The numerical results show good agreement with field theory calculations for a number of impurity configurations and the cross-over function for the magnetic susceptibility from high- to low-temperature behavior shows the expected data collapse. Our results also give quantitative predictions for experiments on doped spin-1/2 chains.
The density matrix renormalization group (DMRG) is one of the most powerful numerical methods available for many-body systems. It has been applied to solve many physical problems, including the calculation of ground states and dynamical properties. In this work, we develop a perturbation theory of the DMRG (PT-DMRG) to greatly increase its accuracy in an extremely simple and efficient way. Using the canonical matrix product state (MPS) representation for the ground state of the considered system, a set of orthogonal basis functions $\left\{|{\ensuremath{\psi}}_{i}\ensuremath{\rangle}\right\}$ is introduced to describe the perturbations to the ground state obtained by the conventional DMRG. The Schmidt numbers of the MPS that are beyond the bond dimension cutoff are used to define these perturbation terms. The perturbed Hamiltonian is then defined as ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}}_{ij}=\ensuremath{\langle}{\ensuremath{\psi}}_{i}|\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}|{\ensuremath{\psi}}_{j}\ensuremath{\rangle}$; its ground state permits us to calculate physical observables with a considerably improved accuracy compared to the original DMRG results. We benchmark the second-order perturbation theory with the help of a one-dimensional Ising chain in a transverse field and the Heisenberg chain, where the precision of the DMRG is shown to be improved O(10)O(10)O(10) times. Furthermore, for moderate LLL the errors of the DMRG and PT-DMRG both scale linearly with ${L}^{\ensuremath{-}1}$ (with LLL being the length of the chain). The linear relation between the dimension cutoff of the DMRG and that of the PT-DMRG at the same precision shows a considerable improvement in efficiency, especially for large dimension cutoffs. In the thermodynamic limit we show that the errors of the PT-DMRG scale with $\sqrt{{L}^{\ensuremath{-}1}}$. Our work suggests an effective way to define the tangent space of the ground-state MPS, which may shed light on the properties beyond the ground state. This second-order PT-DMRG can be readily generalized to higher orders, as well as applied to models in higher dimensions.
The application of real space renormalization group methods to quantum lattice models has become a topic of great interest following the development of the density matrix renormalization group by White.This method has been used to find the ground and low-lying excited state energies and wave functions of quantum spin models in which the form of the ground state is not clear, for instance because the interactions are frustrated.It has also been applied to fermion problems where the tendency for localization due to the strong Coulomb repulsion is opposed by the lowering of the kinetic energy which occurs as a result of electron transfer.The approach is particularly suitable for one-, or quasi-one-dimensional problems.The method involves truncating the Hilbert space in a systematic and optimised manner.Results for the ground state energy are thus variational bounds.The results for low-lying energies and correlation functions for one-dimensional systems have unprecedented accuracy and the method has become the method of choice for solving one-dimensional quantum spin problems.We review the method and results obtained for the spin-1 chain with biquadratic exchange as well as the spin-1/2 model with competing nearest and next nearest neighbour exchange will be described.More recently, the density matrix renormalization group has been applied to reformulate the coupling constant renormalization group approach which is appropriate for the study of critical properties.This approach has been applied to the anisotropic spin-1/2 Heisenberg chain.Finally, we discuss recent work which has borne promising applications in two dimensions -the Ising model and the two-dimensional Hubbard model.
We investigate the application of the density matrix renormalization group (DMRG) to the Hubbard model in momentum space. We treat the one-dimensional models with dispersion relations corresponding to nearest-neighbor hopping and 1/r1/r1/r hopping and the two-dimensional model with isotropic nearest-neighbor hopping. By comparing with the exact solutions for both one-dimensional models and with exact diagonalization in two dimensions, we first investigate the convergence of the ground-state energy. We find variational convergence of the energy with the number of states kept for all models and parameter sets. In contrast to the real-space algorithm, the accuracy becomes rapidly worse with increasing interaction and is not significantly better at half filling. We compare the results for different dispersion relations at fixed interaction strength over bandwidth and find that extending the range of the hopping in one dimension has little effect, but that changing the dimensionality from one to two leads to lower accuracy at weak-to-moderate interaction strength. In the one-dimensional models at half filling, we also investigate the behavior of the single-particle gap, the dispersion of spinon excitations, and the momentum distribution function. For the single-particle gap, we find that proper extrapolation in the number of states kept is important. For the spinon dispersion, we find that good agreement with the exact forms can be achieved at weak coupling if the large momentum-dependent finite-size effects are taken into account for nearest-neighbor hopping. For the momentum distribution, we compare with various weak-coupling and strong-coupling approximations and discuss the importance of finite-size effects as well as the accuracy of the DMRG.