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Acta Physica Sinica

BLOCKSPIN SCHEME AND THE CLUSTER-UPDATING ALGO-RITHMS FOR THE QUANTUM SPIN SYSTEMS

Ying He-Ping, U. J. WIESB

Published Jan 1, 1993• 0 citations
Publisher

Abstract

In this paper, we proposed a blockspin scheme and developed the usual cluster-updating algorithms for the one-and two-dimensional quantum spin systems, saying s=1/2 isotropic Hei-senberg models. The results for he ferro-and antiferro-magnet chains showed that the criti-cal slowing down (CSD) was indeed eliminated, and so it offered a possibility to improve the estimations in the lower-temperature region. We expect that the algorithms could also be ap-plied to the anisotropic or/and the higher-spin cases, as will as some other interesting models, such as eight-vortex lattice model and low-dimensional Fermi-systems.

DOI
10.7498/aps.42.665

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In this paper, we proposed a blockspin scheme and developed the usual cluster-updating algorithms for the one-and two-dimensional quantum spin systems, saying s=1/2 isotropic Hei-senberg models. The results for he ferro-and antiferro-magnet chains showed that the criti-cal slowing down (CSD) was indeed eliminated, and so it offered a possibility to improve the estimations in the lower-temperature region. We expect that the algorithms could also be ap-plied to the anisotropic or/and the higher-spin cases, as will as some other interesting models, such as eight-vortex lattice model and low-dimensional Fermi-systems.
Cluster algorithms are developed for simulating quantum spin systems like the one- and two-dimensional Heisenberg ferro- and anti-ferromagnets. The corresponding two- and three-dimensional classical spin models with four-spin couplings are maped to blockspin models with two-blockspin interactions. Clusters of blockspins are updated collectively. The efficiency of the method is investigated in detail for one-dimensional spin chains. Then in most cases the new algorithms solve the problems of slowing down from which standard algorithms are suffering. Two-dimensional quantum spin systems are relevant for the description of the undoped anti-ferromagnetic precursor insulators of high-Tc superconductors. Presumably understanding the physics of the superconductors requires an understanding of their precursor insulators. This already is a nontrivial problem, which most likely does not allow for a complete analytic solution. Therefore it is natural to use a numerical approach to compute the properties of these materials. At present different
We present a numerical study using a cluster algorithm for the 1-d S=1/2S=1/2S=1/2 quantum Heisenberg models. The dynamical critical exponent for anti-ferromagnetic chains is z=0.0(1)z=0.0(1)z=0.0(1) such that critical slowing down is eliminated.
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.