Research of the Spin Statistical System by the High Order Calculation of the High and Low-Temperature Expansion Series

有限格子方法是生成二维统计系统高低温展开级数的最有效工具。首席研究员H.Arisue通过应用改进的有限格子方法算法，将正方形晶格上XY模型的自由能高温级数从原来的22阶逆温度扩展到48阶。该算法最初是由H.Arisue和他的合作者K.Tabata开发的，用于二维的实体对实体模型。自由能的长序列使我们可以得出结论，当模型的相变是Kosterlitz-Thouless类型时，自由能的行为与预期的高精度是一致的。H.Arisue还用改进的有限格子算法计算了XY模型在正方形晶格上的磁化率和关联函数的二阶、四阶矩的高温序列到33阶。得到的长序列允许我们估计相变的临界点为β=1.1200(1)，这与先前通过蒙特卡罗模拟得到的最精确的值是一致的。

The finite lattice method is the most efficient tool to generate the high- and low-temperature expansion series for the statistical system in two dimensions. The head investigator H. Arisue extended the high-temperature series of the free energy for the XY model on the square lattice to 48th order from the previous 22nd order of the inverse temperature by applying an improved algorithm of the finite lattice method. The algorithm was originally developed by H. Arisue and his collaborator K. Tabata for Solid-on-Solid model in two dimensions. The long series obtained for the free energy allows us to conclude that the behavior of the free energy is consistent to high accuracy with what is expected when the phase transition of the model is of the Kosterlitz-Thouless type. H. Arisue also calculated the high-temperature series of the susceptibility and the second and fourth moments of the correlation function for the XY model on the square lattice to 33rd order by applying the improved algorithm of the finite lattice method. The long series obtained allows us to estimate the critical point of the phase transition as β =1.1200 (1) which is consistent with the most precise value obtained previously by the Monte Carlo simulation.