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Series expansion research of the phase transition for the spin statistical systems

自旋统计系统相变的级数展开研究

基本信息

批准号：
12640382
负责人：
ARISUE Hiroaki
金额：
$ 2.37万
依托单位：
Osaka Prefectural College of Technololy
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640382/
关键词：
phase transition Ising model high-temperature expansion finite lattice method critical exponent specific heat magnetic susceptibility correlation length 自由エネルギー

项目摘要

The finite lattice method is the most efficient method to generate the high-and low-temperature expansion series for the statistical system in two dimensions, but it had not been overwhelming the diagramatic method in three dimensions. In 2000 the head investigator H. Arisue and his collaborator T. Fujiwara (Professor of Kitasato University) developed a new algorithm of the finite lattice method, which enables us to generate the high-temperature series for the Ising model in three dimensions in much higher order than the previous finite lattice method and the diagramatic method. In fact we extended the high-temperature series for the specific heat of the Ising model in three dimensions from the previous 26th order to 46th order in the inverse temperature using the new algorithm in 2000 and 2001 and to 50th order in 2002. We also extended the high-temperature series for the magnetic susceptibility of the same model to 32nd order by the new algorithm from 25th order given by the diagramatic method. In 2003 we applied the algorithm to the high-temperature expansion of the second moment correlation length to 32nd order. These calculations were done partly by the workstation introduced in 2000 and partly by the calculation server at Yukawa Institute for Fundamental Physics, Kyoto University and the massively parallel supercomputer CP-PACS, Tuskuba University. The obtained series for the second moment correlation length were combined with the series for the magnetic susceptibility, giving gave the estimate of the phase transition point of the model in the accuracy of 7. digits, which coincides the estimate by the recent large-scaled numerical simulation, and also giving the estimate for the critical exponents of the magnetic susceptibility and the correlation length in the accuracy of 5 digits, which is the most precise when compared with the estimate by the various other methods.

有限格点方法是生成二维统计系统高低温展开式的最有效方法，但在三维空间中还没有压倒热力学方法。2000年，首席研究员H. Arisue和他的合作者T. Fujiwara（北里大学教授）开发了一种新的有限格点方法算法，该算法使我们能够以比以前的有限格点方法和数值方法高得多的阶数生成三维Ising模型的高温序列。事实上，我们延长了高温系列的比热的伊辛模型在三维从以前的第26阶到第46阶的逆温度在2000年和2001年使用新的算法和第50阶在2002年。我们还用新算法将同一模型的高温磁化率序列从解析法给出的25阶扩展到32阶。2003年我们将该算法应用于二阶矩关联长度的高温扩展到32阶。这些计算部分由2000年引入的工作站完成，部分由京都大学汤川基础物理研究所的计算服务器和Tuskuba大学的大规模并行超级计算机CP-PACS完成。将得到的二阶矩关联长度序列与磁化率序列相结合，给出了模型相变点的估计，精度为7。给出了磁化率临界指数和相关长度的5位精度估计，与其它方法的估计结果相比，该估计精度最高。