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Progress of Theoretical Physics

An Improved Transfer Matrix Method for Quantum Spin Systems. II: Isotropic Heisenberg Chain with Spin 1/2

T. Tsuzuki

Published Feb 1, 1986• 11 citations
Publisher

Abstract

An improved transfer matrix method which has been proposed by the author is applied to the isotropic Heisenberg chain with spin 1/2. Both ferro- and antiferromagnetic couplings are considered. We first explain how to find the size of transfer matrix in consideration of the symmetry of Trotter subsystem with Trotter number M. We then compute the internal energy, specific heat, magnetization and susceptibility in the presence as well as absence of magnetic field in the cases of the pair and tri spin decompositions with M=2, 4 and 6. Our results are qualitatively explained on the basis of the level schemes of the clusters of decomposition. We find good and rapid convergence to the infinite chain limit of Bonner and Fisher with increasing Trotter number in the tri spin decomposition except at very low temperatures, when we examine the zero field limit. Convergence with Trotter number in the case of finite field is also rapid in either decomposition. However there are interesting discrepancies, though small, between the results of Bonner and Fisher and ours in the tri spin decomposition, which remain as one of future problems to be investigated. Finally we present brief discussion about strategy and tactics for getting improved results by our method.

DOI
10.1143/ptp.75.225

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Acta Physica Polonica A•Aug 1, 1997•1 citation
An improved transfer matrix method which has been proposed by the author is applied to the isotropic Heisenberg chain with spin 1/2. Both ferro- and antiferromagnetic couplings are considered. We first explain how to find the size of transfer matrix in consideration of the symmetry of Trotter subsystem with Trotter number M. We then compute the internal energy, specific heat, magnetization and susceptibility in the presence as well as absence of magnetic field in the cases of the pair and tri spin decompositions with M=2, 4 and 6. Our results are qualitatively explained on the basis of the level schemes of the clusters of decomposition. We find good and rapid convergence to the infinite chain limit of Bonner and Fisher with increasing Trotter number in the tri spin decomposition except at very low temperatures, when we examine the zero field limit. Convergence with Trotter number in the case of finite field is also rapid in either decomposition. However there are interesting discrepancies, though small, between the results of Bonner and Fisher and ours in the tri spin decomposition, which remain as one of future problems to be investigated. Finally we present brief discussion about strategy and tactics for getting improved results by our method.
An improved transfer matrix method is proposed for calculating the partition function of quantum spin systems, which is expressed by the M-th approximant of the Suzuki-Trotter transformation with Trotter number M. The point is to incorporate the symmetry property of the Trotter subsystem in calculation. The size of our transfer matrix is reduced very much. Thermodynamics is described by the maximum eigenvalue of the transfer matrix. Our method is applied to the XY model with spin 1/2. We consider the cases of pair and tri spin decompositions with Trotter number up to 4 (M≤4). We can do our calculation analytically in those cases. In the pair spin decomposition we find results which have been obtained by the Monte Carlo simulation and by a transfer matrix method by computer. We find nice results in the tri spin decomposition. Our result in the case M=4 is as good as the result in M=8 by the previous Monte Carlo simulation on comparing them with the exact result of Katsura.
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.
The quantum transfer-matrix method (QTM) has been applied to study the finite-temperature static properties of the spin S=1S=1S=1 antiferromagnetic Heisenberg chains in a wide range of the single-ion anisotropy and temperatures. The high-resolution QTM simulation data are obtained for the zero-field susceptibility, specific heat, as well as for the field-dependent magnetization. The microscopic parameters of a number of real quasi-one-dimensional compounds are found from fitting procedures, some theoretical approaches are numerically verified and low-temperature extensions of high-temperature series are given in terms of Chebyshev polynomials.
A formulation is given to map the one-dimensional S = 1/2 quantum spin systems onto the two-dimensional Ising model with many-spin interactions on the basis of the n-spin cluster decomposition (n CLD). The cluster transfer matrix is defined which characterizes the thermodynamical properties of the equivalent system. A prescription is presented to construct the cluster transfar matrix and to evaluate its maximum eigenvalue numerically. The method has been applied to the calculation of the energy and specific heat of the XY model for various values of the cluster size n and th Trotter size m. The convergence properties of the energy as a function of n and m have been studied in detail. The main results obtained are: (1) The low-temperature behavior is improved much more by considering larger spin clusters; (2) the energy depends on the spin cluster through the effective cluster size ne defined by ne =(n+1)/2; and (3) the energy converges linearly with 1/n2em2 except at very low temperatures.
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.