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arXiv:cond-mat/0203500 · Physical review. B, Condensed matter

Dynamical density-matrix renormalization-group method

Eric Jeckelmann

Published Jul 26, 2002• 374 citations
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Abstract

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.

DOI
10.1103/physrevb.66.045114

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

Robert J. Bursill

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.

arXiv:cond-mat/0103186•Physical review. B, Condensed matter•Mar 30, 2001•7 citations
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.
The calculation of dynamic correlation functions in quantum systems is formulated as a variational problem. For low-dimensional many-body systems this variational problem can be solved numerically using the density-matrix renormalisation group (DMRG). This dynamic DMRG method is demonstrated on the linear optical conductivity in the Mott insulating phase of the one-dimensional extended Hubbard model at half filling. The full optical spectrum of this model can be calculated almost exactly for chains with more than 100 sites, which is large enough to investigate the spectral properties in the thermodynamic limit. The accuracy of the method is illustrated by comparison with analytical results in the field-theoretical regime and in the strong-coupling limit.
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.
The density matrix renormalization group (DMRG) method and its applications to finite temperatures and two-dimensional systems are reviewed. The basic idea of the original DMRG method, which allows precise study of the ground-state properties and low-energy excitations, is presented for models which include long-range interactions. The DMRG scheme is then applied to the diagonalization of the quantum transfer matrix for one-dimensional systems, and a reliable algorithm at finite temperatures is formulated. Dynamic correlation functions at finite temperatures are calculated from the eigenvectors of the quantum transfer matrix with analytical continuation to the real frequency axis. An application of the DMRG method to two-dimensional quantum systems in a magnetic field is demonstrated and reliable results for quantum Hall systems are presented.
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.
The density-matrix renormalization-group (DMRG) method is used to investigate optical excitations in the Mott insulating phase of a one-dimensional extended Hubbard model. The linear optical conductivity is calculated using the dynamical DMRG method and the nature of the lowest optically excited states is investigated using a symmetrized DMRG approach. The numerical calculations agree perfectly with field-theoretical predictions for a small Mott gap and analytical results for a large Mott gap obtained with a strong-coupling analysis. It is shown that four types of optical excitations exist in this Mott insulator: pairs of unbound charge excitations, excitons, excitonic strings, and charge-density-wave (CDW) droplets. Each type of excitation dominates the low-energy optical spectrum in some region of the interaction parameter space and corresponds to distinct spectral features: a continuum starting at the Mott gap (unbound charge excitations), a single peak or several isolated peaks below the Mott gap (excitons and excitonic strings, respectively), and a continuum below the Mott gap (CDW droplets).
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.