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

有限点阵法是生成二维统计系统的高低温展开级数最有效的方法，但在三维中还没有压倒图解法。2000年，首席研究员H. Arisue和他的合作者T. Fujiwara（北中大学教授）开发了一种新的有限晶格法算法，使我们能够以比以前的有限晶格法和图解法高得多的顺序在三维中生成Ising模型的高温序列。实际上，我们在2000年和2001年使用新算法将三维Ising模型比热的高温级数从以前的26阶扩展到逆温度的46阶，并在2002年扩展到50阶。我们还用新算法将同一模型磁化率的高温级数由图解法给出的25阶扩展到32阶。2003年，我们将该算法应用于二阶矩相关长度的高温展开到32阶。这些计算部分是由2000年引进的工作站完成的，部分是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完成的。将得到的二阶矩相关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。数字，这与最近大规模数值模拟的估计相吻合，并给出了5位数精度的磁化率和相关长度的临界指数的估计，与其他各种方法的估计相比，这是最精确的。

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.