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arXiv:cond-mat/9503107 · Physical Review Letters

Thermodynamic Limit of Density Matrix Renormalization

Stellan S. Ostlund, Stefan Rommer

Published Nov 6, 1995• 1,077 citations
arXivHTMLPDF

Abstract

The density matrix renormalization group discovered by White is investigated. In the case where renormalization eventually converges to a fixed point we show that quantum states in the thermodynamic limit with periodic boundary conditions can be simply represented by a ``matrix product ground state'' with a natural description of Bloch states of elementary excitations. We then observe that these states can be rederived through a simple variational ansatz making no reference to a renormalization construction. The method is tested on the spin-1 Heisenberg model.

DOI
10.1103/physrevlett.75.3537

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The density matrix renormalization group discovered by White is investigated. In the case where renormalization eventually converges to a fixed point we show that quantum states in the thermodynamic limit with periodic boundary conditions can be simply represented by a ``matrix product ground state'' with a natural description of Bloch states of elementary excitations. We then observe that these states can be rederived through a simple variational ansatz making no reference to a renormalization construction. The method is tested on the spin-1 Heisenberg model.
We investigate the density matrix renormalization group (DMRG) discovered by White and show that in the case where the renormalization eventually converges to a fixed point the DMRG ground state can be simply written as a matrix-product form. This ground state can also be rederived through a simple variational ansatz making no reference to the DMRG construction. We also show how to construct the matrix-product states and how to calculate their properties, including the excitation spectrum. This paper provides details of many results announced earlier.
We investigate convergence of the density-matrix renormalization group (DMRG) in the thermodynamic limit for gapless systems. Although the DMRG correlations always decay exponentially in the thermodynamic limit, the correlation length at the DMRG fixed-point scales as $\ensuremath{\xi}\ensuremath{\sim}{m}^{1.3},$ where m is the number of kept states, indicating the existence of algebraic order for the exact system. The single-particle excitation spectrum is calculated, using a Bloch-wave ansatz, and we prove that the Bloch-wave ansatz leads to the symmetry $E(k)=E(\ensuremath{\pi}\ensuremath{-}k)$ for translationally invariant half-integer spin-systems with local interactions. Finally, we provide a method to compute overlaps between ground states obtained from different DMRG calculations.
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.
We propose a formalism to study dynamical properties of a quantum many-body system in the thermodynamic limit by studying a finite system with ``infinite boundary conditions'' where both finite-size effects and boundary effects have been eliminated. For one-dimensional systems, infinite boundary conditions are obtained by attaching two boundary sites to a finite system, where each of these two sites effectively represents a semi-infinite extension of the system. One can then use standard finite-size matrix product state techniques to study a region of the system while avoiding many of the complications normally associated with finite-size calculations such as boundary Friedel oscillations. We illustrate the technique with an example of time evolution of a local perturbation applied to an infinite (translationally invariant) ground state, and use this to calculate the spectral function of the S=1S=1S=1 Heisenberg spin chain. This approach is more efficient and more accurate than conventional simulations based on finite-size matrix product state and density-matrix renormalization-group approaches.
We describe how to implement the time-dependent variational principle for matrix product states in the thermodynamic limit for nonuniform lattice systems. This is achieved by confining the nonuniformity to a (dynamically expandable) finite region with fixed boundary conditions. The suppression of nonphysical quasiparticle reflections from the boundary of the nonuniform region is also discussed. Using this algorithm we study the dynamics of localized excitations in infinite systems, which we illustrate in the case of the spin-1 antiferromagnetic Heisenberg model and the ${\ensuremath{\phi}}^{4}$ model.
We present a real-space renormalization group approach for the corner Hamiltonian, which is relevant to the reduced density matrix in the density matrix renormalization group. A set of self-consistent equations that the renormalized Hamiltonian should satisfy in the thermodynamic limit is also derived from the fixed point of the recursion relation for the corner Hamiltonian. We demonstrate the renormalization group algorithm for the S=1/2S=1/2S=1/2 XXZ spin chain and show that the results are consistent with the exact solution. We further examine the renormalization group for the S=1 Heisenberg spin chain and then discuss the nature of the eigenvalue spectrum of the corner Hamiltonian for the non-integrable model.