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arXiv:cond-mat/9706152 · Journal of the Physical Society of Japan

Thermodynamics of the Anisotropic Heisenberg Chain Calculated by the Density Matrix Renormalization Group Method

Naokazu Shibata

Published Aug 15, 1997• 149 citations
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Abstract

The density matrix renormalization group (DMRG) method is applied to the anisotropic Heisenberg chain at finite temperatures. The free energy of the system is obtained using the quantum transfer matrix which is iteratively enlarged in the imaginary time direction. The magnetic susceptibility and the specific heat are calculated down to T=0.01J and compared with the Bethe ansatz results. The agreement including the logarithmic correction in the magnetic susceptibility at the isotropic point is fairly good.

DOI
10.1143/jpsj.66.2221

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The density matrix renormalization group (DMRG) method is applied to the anisotropic Heisenberg chain at finite temperatures. The free energy of the system is obtained using the quantum transfer matrix which is iteratively enlarged in the imaginary time direction. The magnetic susceptibility and the specific heat are calculated down to T=0.01J and compared with the Bethe ansatz results. The agreement including the logarithmic correction in the magnetic susceptibility at the isotropic point is fairly good.
Thermodynamic properties of the quantum Heisenberg spin chains with S=1/2,S=1/2,S=1/2, 1, and 3/23/23/2 are investigated using the transfer-matrix renormalization-group method. The temperature dependence of the magnetization, susceptibility, specific heat, spin-spin correlation length, and several other physical quantities in a zero or finite applied field are calculated and compared. Our data agree well with the Bethe ansatz, exact diagonalization, and quantum Monte Carlo results and provide further insight into the quantum effects in the antiferromagnetic Heisenberg spin chains.
It is shown that White's density matrix renormalization group technique can be adapted to obtain thermodynamic quantities. As an illustration, the magnetic susceptibility of Heisenberg S=1/2S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2S=1/2 and S=3/2S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3/2S=3/2 spin chains are computed. A careful finite size analysis is made to determine the range of temperatures where the results are reliable. For the S=1/2S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2S=1/2 chain, the comparison with the exact Bethe ansatz curve shows an agreement within 11%1 down to T=0.05JT\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.05JT=0.05J.
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.
By extending White's density-matrix renormalization group (DMRG) technique to finite temperatures, we calculate the impurity susceptibility for the following systems: (a) a spin-1/2 impurity coupled to a single site of an antiferromagnetic Heisenberg spin-1/2 chain and (b) an impurity coupled symmetrically to the two spins at the open ends of the chain. We also perform a Monte Carlo simulation and show that the DMRG adaptation gives reliable results. The calculated impurity susceptibility shows Pauli-like behavior at low temperatures in the former case while a ln T divergence is found in the latter case, in agreement with predictions based on conformal field theory.
The density-matrix renormalization group (DMRG) applied to transfer matrices allows it to calculate static as well as dynamical properties of one-dimensional quantum systems at finite temperature in the thermodynamic limit. To this end the quantum system is mapped onto a two-dimensional classical system by a Trotter-Suzuki decomposition. Here we discuss two different mappings: The standard mapping onto a two-dimensional lattice with checkerboard structure as well as an alternative mapping introduced by two of us. For the classical system an appropriate quantum transfer matrix is defined which is then treated using a DMRG scheme. As applications, the calculation of thermodynamic properties for a spin-1/2 Heisenberg chain in a staggered magnetic field and the calculation of boundary contributions for open spin chains are discussed. Finally, we show how to obtain real time dynamics from a classical system with complex Boltzmann weights and present results for the autocorrelation function of the XXZ-chain.