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Communications in Theoretical Physics

An Efficient Cluster Algorithm and Its Numerical Study for the Quantum Spin systems

He-Ping Ying, JI DA-REN

Published Jun 15, 1994• 0 citations
Publisher

Abstract

Using a blockspin scheme, a cluster algorithm is developed for the numerical study of the quantum spin systems. The method is tested in detail for the ferro- and antiferromagnetic Heisenberg chains, of which our calculations show that the new algorithm can be used to simulate the induced classical spin systems efficiently by strongly reducing the slowing down effects.

DOI
10.1088/0253-6102/21/4/421

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Using a blockspin scheme, a cluster algorithm is developed for the numerical study of the quantum spin systems. The method is tested in detail for the ferro- and antiferromagnetic Heisenberg chains, of which our calculations show that the new algorithm can be used to simulate the induced classical spin systems efficiently by strongly reducing the slowing down effects.
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
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.
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.
We apply a meron cluster algorithm to the XY spin chain, which describes a quantum rotor. This is a multi-cluster simulation supplemented by an improved estimator, which deals with objects of half-integer topological charge. This method is powerful enough to provide precise results for the model with a θ-term — it is therefore one of the rare examples, where a system with a complex action can be solved numerically. In particular we measure the correlation length, as well as the topological and magnetic susceptibility. We discuss the algorithmic efficiency in view of the critical slowing down. Due to the excellent performance that we observe, it is strongly motivated to work on new applications of meron cluster algorithms in higher dimensions.