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arXiv:cond-mat/9812349 · Physical review. B, Condensed matter

Density-matrix renormalization-group algorithm for quantum lattice systems with a large number of states per site

Robert J. Bursill

Published Jul 15, 1999• 31 citations
arXivHTMLPDF

Abstract

A variant of White's density-matrix renormalization-group scheme, which is designed to compute low-lying energies of one-dimensional quantum lattice models with a large number of degrees of freedom per site is described. The method is tested on two exactly solvable models---the spin-$\frac{1}{2}$ antiferromagnetic Heisenberg chain and a dimerized $\mathrm{XY}$ spin chain. To illustrate the potential of the method, it is applied to a model of spins interacting with quantum phonons. It is shown that the method accurately resolves a number of energy gaps on periodic rings that are sufficiently large to afford an accurate investigation of critical properties via the use of finite-size scaling theory.

DOI
10.1103/physrevb.60.1643

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A variant of White's density-matrix renormalization-group scheme, which is designed to compute low-lying energies of one-dimensional quantum lattice models with a large number of degrees of freedom per site is described. The method is tested on two exactly solvable models---the spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg chain and a dimerized XY\mathrm{XY}XY spin chain. To illustrate the potential of the method, it is applied to a model of spins interacting with quantum phonons. It is shown that the method accurately resolves a number of energy gaps on periodic rings that are sufficiently large to afford an accurate investigation of critical properties via the use of finite-size scaling theory.
The density matrix renormalization group (DMRG) method generates the low-energy states of linear systems of NNN sites with a few degrees of freedom at each site by starting with a small system and adding sites step by step while keeping constant the dimension of the truncated Hilbert space. DMRG algorithms are adapted to open chains with inversion symmetry at the central site, to cyclic chains and to weakly coupled chains. Physical properties rather than energy accuracy is the motivation. The algorithms are applied to the edge states of linear Heisenberg antiferromagnets with spin S≥1S \ge 1S≥1 and to the quantum phases of a frustrated spin-1/2 chain with exchange between first and second neighbors. The algorithms are found to be accurate for extended Hubbard and related 1D models with charge and spin degrees of freedom.
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.
A formulation of numerical real-space renormalization groups for quantum many-body problems is presented and several algorithms utilizing this formulation are outlined. The methods are presented and demonstrated using S=1/2 and S=1 Heisenberg chains as test cases. The key idea of the formulation is that rather than keep the lowest-lying eigenstates of the Hamiltonian in forming a new effective Hamiltonian of a block of sites, one should keep the most significant eigenstates of the block density matrix, obtained from diagonalizing the Hamiltonian of a larger section of the lattice which includes the block. This approach is much more accurate than the standard approach; for example, energies for the S=1 Heisenberg chain can be obtained to an accuracy of at least ${10}^{\mathrm{\ensuremath{-}}9}$. The method can be applied to almost any one-dimensional quantum lattice system, and can provide a wide variety of static properties.
We present a simple method, combining the density-matrix renormalization-group algorithm with finite-size scaling, which permits the study of critical behavior in quantum spin chains. Spin moments and dimerization are induced by boundary conditions at the chain ends and these exhibit power-law decay at critical points. Results are presented for the spin-1/21/21/2 Heisenberg antiferromagnet; an analytic calculation shows that logarithmic corrections to scaling can sometimes be avoided. We also examine the spin-1 chain at the critical point separating the Haldane gap and dimerized phases. Exponents for the dimer-dimer and the spin-spin correlation functions are consistent with results obtained from bosonization.
In this thesis we discuss in detail the density-matrix renormalization group (DMRG) for simulating low-energy properties of quantum spin models. We implement an original DMRG routine on the S=1/2 antiferromagnetic Heisenberg chain and benchmark its efficiency against exact results (energies, correlation functions, etc.) as well as conformal field-theoretical calculations due to finite-size scaling (ground-state energy and spin gap logarithmic corrections). Moreover, we apply the DMRG to a two-leg square ladder system, where in addition to bilinear exchange terms, we also consider an additional cyclic four-spin ring-exchange. The transposition of four spins gives rise to biquadratic exchange terms which are non-trivial to implement in the DMRG. Intermediate results of the ring-exchange are presented along with the difficulties presently encountered.
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.