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Journal of the Physical Society of Japan

The Cluster Decomposition Approach to Finite Quantum Spin Systems

Hidetsugu Kitatani, Takehiko Oguchi

Published Apr 15, 1987• 1 citation
Publisher

Abstract

A new method is presented which can deal with large finite quantum spin systems (which are so large that direct diagonalization is not possible) by calculating exactly Trotter formula for each finite Trotter number n , and extrapolating them to infinite n . This method is powerful for calculating spin systems which contain frustration. A simple example is also discussed.

DOI
10.1143/jpsj.56.1320

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A new method is presented which can deal with large finite quantum spin systems (which are so large that direct diagonalization is not possible) by calculating exactly Trotter formula for each finite Trotter number n , and extrapolating them to infinite n . This method is powerful for calculating spin systems which contain frustration. A simple example is also discussed.
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.
A simple Monte Carlo high-temperature expansion method to evaluate the partition function of quantum spin systems which was proposed in the previous paper l) is improved so that all of the needed expansion-coefficients are obtained from interpolating points of less than ten with an interpolation formula. By this improvement, this method is applicable to the finite spin-systems of rather large spin size regardless of dimensionality of the system. As an example, the method is applied to the one-dimensional spin-l/2 isotropic Heisenberg ferromagnetic system.
I present a new and efficient disconnected cluster decomposition to perform Monte Carlo Simulations based on the Trotter-Suzuki method for 1-d quantum systems. As opposed to the normal checkerboard world line algorithm this mapping allows one to simulate models with interactions up to next nearest neighbours or even one-dimensional double layers. I have used this cluster decomposition to simulate the one-dimensional t-J-J' model at α=J'/J=0.5 and have investigated the sign problem in this model. At half band filling the system is dimerised into singlet spin pairs when α=0.5 due to frustration. I believe that the spin gap is no longer present around quarter band filling
We have developed a technique to replace hashing in implementing the Lancz\"os method for exact diagonalization of quantum-spin models that enables us to carry out numerical studies on substantially larger lattices than previously studied. We describe the algorithm in detail and present results for the ground-state energy, the first-excited-state energy, and the spin-spin correlations on various finite lattices for spins S=1/2, 1, 3/2, and 2. Results for an infinite system are obtained by extrapolation. We also discuss the generalization of our method to other models.