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Physical review. B, Condensed matter

Density-matrix renormalization-group method for excited states

Michael Chandross, J. Charles Hicks

Published Apr 15, 1999• 13 citations
Publisher

Abstract

The density-matrix renormalization-group (DMRG) method can provide approximate ground-state wave functions for many one-dimensional systems with a high degree of accuracy. Conventional DMRG techniques, however, cannot be expected to give accurate excited-state information because the excited states of interest are not explicitly included in the calculations. We discuss techniques for extending DMRG to accurately target and investigate arbitrary excited states. As an example we apply the method to investigate the properties of excitons within the extended Hubbard model. Our results show that excited-state calculations within the standard DMRG procedure give results that are quantitatively incorrect.

DOI
10.1103/physrevb.59.9699

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Comment on “Density-matrix renormalization-group method for excited states”

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In a recent paper [Phys. Rev. B $59,$ 9699 (1999)], Chandross and Hicks claim to present a new density-matrix renormalization group (DMRG) method for dealing with excited-states of quantum lattice models. The proposed improvement to the DMRG---the inclusion of excited-state wave functions in addition to the ground state in the density matrix when calculating excitations---is in fact standard practice, is clearly stated in White's original papers, and has been used repeatedly by many groups to study excited states. The authors apply the method to the extended, dimerized Hubbard model for conjugated polymers. The criteria for determining whether states are bound or not are assessed. The authors claim that their results show that the optically important $``{1B}_{u}''$ state is bound (excitonic), in contrast to a previous study. However, the discussion is qualitative, and the authors arrive at conclusions on the basis of results for one lattice size only. We show that when the criterion of Chandross and Hicks is developed into a quantitative definition of particle-hole separation, with the finite-size dependence analyzed, the implication is that the ${1B}_{u}$ state is unbound, at least in the sense of the density-density correlation function, in keeping with the conclusions of a previous study.

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•Jan 1, 1998•0 citations
The density-matrix renormalization-group (DMRG) method can provide approximate ground-state wave functions for many one-dimensional systems with a high degree of accuracy. Conventional DMRG techniques, however, cannot be expected to give accurate excited-state information because the excited states of interest are not explicitly included in the calculations. We discuss techniques for extending DMRG to accurately target and investigate arbitrary excited states. As an example we apply the method to investigate the properties of excitons within the extended Hubbard model. Our results show that excited-state calculations within the standard DMRG procedure give results that are quantitatively incorrect.
In a recent paper [Phys. Rev. B 59,59,59, 9699 (1999)], Chandross and Hicks claim to present a new density-matrix renormalization group (DMRG) method for dealing with excited-states of quantum lattice models. The proposed improvement to the DMRG---the inclusion of excited-state wave functions in addition to the ground state in the density matrix when calculating excitations---is in fact standard practice, is clearly stated in White's original papers, and has been used repeatedly by many groups to study excited states. The authors apply the method to the extended, dimerized Hubbard model for conjugated polymers. The criteria for determining whether states are bound or not are assessed. The authors claim that their results show that the optically important ‘‘1Bu′′``{1B}_{u}''‘‘1Bu′′​ state is bound (excitonic), in contrast to a previous study. However, the discussion is qualitative, and the authors arrive at conclusions on the basis of results for one lattice size only. We show that when the criterion of Chandross and Hicks is developed into a quantitative definition of particle-hole separation, with the finite-size dependence analyzed, the implication is that the 1Bu{1B}_{u}1Bu​ state is unbound, at least in the sense of the density-density correlation function, in keeping with the conclusions of a previous study.
A density-matrix renormalization-group (DMRG) method for calculating dynamical properties and excited states in low-dimensional lattice quantum many-body systems is presented. The method is based on an exact variational principle for dynamical correlation functions and the excited states contributing to them. This dynamical DMRG is an alternate formulation of the correction vector DMRG but is both simpler and more accurate. The finite-size scaling of spectral functions is discussed and a method for analyzing the scaling of dense spectra is described. The key idea of the method is a size-dependent broadening of the spectrum. The dynamical DMRG and the finite-size scaling analysis are demonstrated on the optical conductivity of the one-dimensional Peierls-Hubbard model. Comparisons with analytical results show that the spectral functions of infinite systems can be reproduced almost exactly with these techniques. The optical conductivity of the Mott-Peierls insulator is investigated and it is shown that its spectrum is qualitatively different from the simple spectra observed in Peierls (band) insulators and one-dimensional Mott-Hubbard insulators.
In order to extend the density-matrix renormalization-group (DMRG) method to two-dimensional systems, we formulate two alternative methods to prepare the initial states. We find that the number of states that is needed for accurate energy calculations grows exponentially with the linear system size. We also analyze how the states kept in the DMRG method manage to preserve both the intrablock and interblock Hamiltonians, which is the key to the high accuracy of the method. We also prove that the energy calculated on a finite cluster is always a variational upper bound.
The density matrix renormalization group (DMRG) method allows for very precise calculations of ground state properties in low-dimensional strongly correlated systems. We investigate two methods to expand the DMRG to calculations of dynamical properties. In the Lanczos vector method the DMRG basis is optimized to represent Lanczos vectors, which are then used to calculate the spectra. This method is fast and relatively easy to implement, but the accuracy at higher frequencies is limited. Alternatively, one can optimize the basis to represent a correction vector for a particular frequency. The correction vectors can be used to calculate the dynamical correlation functions at these frequencies with high accuracy. By separately calculating correction vectors at different frequencies, the dynamical correlation functions can be interpolated and pieced together from these results. For systems with open boundaries we discuss how to construct operators for specific wave vectors using filter functions.
In order to extend the density-matrix renormalization-group (DMRG) method to two-dimensional systems, we formulate two alternative methods to prepare the initial states. We find that the number of states that are needed for accurate energy calculations grows exponentially with the linear system size. We also analyze how the states kept in the DMRG method manage to preserve both the intrablock and interblock Hamiltonians, which is the key to the high accuracy of the method. We also prove that the energy calculated on a finite cluster is always a variational upper bound.
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.