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

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工具。首席调查员H。Arisue应用有限格点法的改进算法，将正方格点上XY模型的高温自由能序列从原来的22阶逆温度序列扩展到48阶。该算法最初由H.有休和他的合作者K.二维立体模型的Tabata。获得的自由能的长序列使我们能够得出结论，自由能的行为是一致的，以高精度与预期的模型时的相变是Kosterlitz无相变类型。H. Arisue还应用有限格点法的改进算法计算了XY模型在33阶正方格点上的高温磁化率和相关函数的二阶和四阶矩。得到的长序列使我们能够估计相变的临界点为β =1.1200（1），这与以前通过Monte Carlo模拟得到的最精确的值一致。

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.