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arXiv:cond-mat/9503094 · Physical review. B, Condensed matter

Density-matrix algorithm for the calculation of dynamical properties of low-dimensional systems

K. Hallberg

Published Oct 1, 1995• 191 citations
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Abstract

I extend the scope of the density-matrix renormalization-group technique developed by White to the calculation of dynamical correlation functions using the continuous fraction expansion of the Green's function. As an application and performance evaluation I calculate the spin dynamics of the one-dimensional S=1/2 Heisenberg chain.
DOI
10.1103/physrevb.52.r9827

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We study the one-dimensional S = 1/2 Heisenberg model with an uniform and a staggered magnetic fields, using the dynamical density-matrix renormalization group (DDMRG) technique. The DDMRG enables us to investigate the dynamical properties of chain with lengths up to a few hundreds, and the results are numerically exact in the same sense as 'exact diagonalization' results are. Thus, we can analyze the low-energy spectrum almost in the thermodynamic limit. In this work, we calculate the dynamical spin structure factor and demonstrate the performance of the DDMRG method applying the open-end boundary conditions as well as the periodic boundary conditions.
A numerical approach to ground-state dynamical correlation functions from the density-matrix renormalization group (DMRG) is developed. Using sum rules, moments of a dynamic correlation function can be calculated with DMRG, and with the moments the dynamical correlation function can be obtained by the maximum entropy method. We apply this method to a one-dimensional spinless fermion system, which can be converted to the spin-1/2 Heisenberg model in a special case. The dynamical density-density correlation function is obtained. \textcopyright{} 1996 The American Physical Society.
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) 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.
We use the density-matrix renormalization-group technique developed by White to calculate the spin correlation functions 〈Sn+lz{\mathit{S}}_{\mathit{n}+\mathit{l}}^{\mathit{z}}Sn+lz​ Snz{\mathit{S}}_{\mathit{n}}^{\mathit{z}}Snz​〉=(-1)l{)}^{\mathit{l}})l\ensuremath{\omega}(l,N) for isotropic Heisenberg rings up to N=70 sites. The correlation functions for large l and N are found to obey the scaling relation \ensuremath{\omega}(l,N)=\ensuremath{\omega}(l,\ensuremath{\infty})${\mathit{f}}_{\mathit{XY}}^{\mathrm{\ensuremath{\alpha}}}$(l/N) proposed by Kaplan et al., which is used to determine \ensuremath{\omega}(l,\ensuremath{\infty}). The asymptotic correlation function \ensuremath{\omega}(l,\ensuremath{\infty}) and the magnetic structure factor S(q=\ensuremath{\pi}) show logarithmic corrections consistent with \ensuremath{\omega}(l,\ensuremath{\infty})\ensuremath{\sim}a\ensuremath{\surd}lncl /l, where c is related to the cut-off dependent coupling constant geff{\mathit{g}}_{\mathrm{eff}}geff​(l0{\mathit{l}}_{0}l0​)=1/ln(cl0{\mathit{cl}}_{0}cl0​), as predicted by field theoretical treatments.