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Progress of Theoretical Physics

Study of One-Dimensional Quantum Spin Systems by the Transfer-Matrix Method

Hiroshi Betsuyaku

Published Feb 1, 1985• 68 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.73.319

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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.
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.
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.
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.
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) 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.
The partition function of a quantal spin system is expressed by that of the Ising model, on the basis of the generalized Trotter formula. Thereby the ground state of the d-dimensional Ising model with a transverse field is proven to be equivalent to the (d+1)-dimensional Ising model at finite temperatures. A general relationship is established between the two partition functions of a general quantal spin system and the corresponding Ising model with many-spin interactions, which yields some rigorous results on quantum systems. Some applications are given.