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Progress of Theoretical Physics

An Improved Transfer Matrix Method for Quantum Spin Systems

T. Tsuzuki

Published Jun 1, 1985• 23 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.73.1352

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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.
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.
The Suzuki-Trotter formula has been used to get the mmm th approximant to the classical representation of the partition function of the NNN-spin S=12S=\frac{1}{2}S=21​ XY\mathrm{XY}XY chain. The equivalent two-dimensional ($N\ifmmode\times\else\texttimes\fi{}2m$) Ising model with four-spin interactions has been studied in detail by use of the numerically exact transfer-matrix method for T> 0.05T>~0.05T> 0.05 and m< 8m<~8m< 8. The limiting $m\ensuremath{\rightarrow}\ensuremath{\infty}$ behavior has been estimated from the extrapolation formula of the form $E(m)=E(\ensuremath{\infty})+\frac{a}{{m}^{2}}$. The limiting value of the energy agrees with the exact solution of Katsura.
The Suzuki-Trotter formula has been used to get the mth approximant to the classical representation of the partition function of the one-dimensional N-spin S=½ quantum spin systems. The equivalent two-dimensional (N×2m) Ising model with four-spin interactions has been studied in detail by using the numerically exact transfer-matrix method for T≥0.05 and m ≤8. The convergence properties have been examined in two different representations; checkerboard decomposition (CBD) and real-space decomposition (RSD). The spin correlation functions in RSD converge much faster than those in CBD. The limiting m→∞ behavior has been estimated from the extrapolation formula of the form: E(m)=E( ∞)+ α/ m2. The limiting values of the energy derived from the nearest-neighbor correlation agree with the correct values excellently.
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.
The one-dimensional S=1 quantum spin system has been mapped onto the equivalent two-dimensional Ising model on the basis of the equivalence theorem of Suzuki. The classical representation of the partition function has been expressed in terms of the maximum eigenvalue of the transfer matrix and evaluated by means of the numerically exact transfer-matrix method. The physical meaning of two kinds of correlation functions in RSD and CBD has been made clear in terms of the concept of the maximum eigenstate. Calculations have been made for both ferromagnetic and antiferromagnetic Heisenberg chains with several single-site anisotropy values, both positive and negative. Good agreements are found with some existing numerical results.