High-and Low-temperature expansion for the spin statistical systems using the finite lattice method

用有限格子方法计算了各种自旋系统的各种参数展开式的高阶项，首次计算了Q=5-50时Q态Potts模型的自由能、磁化强度和磁化率的低温展开式，其中系统在展开式参数上表现出一阶到四十一阶的相变.我们还将有限格子方法应用于同一系统的大Q展开，得到了1/√Q中高达23阶的6阶能量累积量的级数。对级数的分析发现，自由能和潜热的值与临界点的精确解有很高的精度一致，比热的级数即使在Q=5的情况下也给出了几个百分点的收敛结果。其次，我们计算了三维伊辛模型、正方形晶格上绝对值SOS模型和三角形晶格上绝对值SOS模型和离散高斯模型的三种界面宽度的低温展开级数，分别到相应模型展开参数的第17、23、22、24和22阶。假设重整化群参数所预测的每个量的临界指数，我们从级数中得到每个模型的粗化转变点的值。它们与前三种模型的蒙特卡罗模拟结果吻合较好，并对后两种模型给出了新的预测。

Using the finite lattice method we calculated the high order terms of the various types of the parameter expansions for various kinds of spin systems.We first calculated the low-temperature expansion series of the free energy, the magnetization and the magnetic susceptibility in the q-state Potts model in two dimensions for all of q=5-50, where the system exhibits first order phase transition to the 41st order in the expansion, parameter. We also applied the finite lattice method to the large-q expansion of the same system and obtained the series for the up to 6th order energy cumulants to the 23rd order in 1/√q. Analyzing the series, we found that the values of the free energy and the latent heat coincide in very high precision with those of the exact solutions at the critical point and the series for the specific heat gives converging results in the accuracy of a few percent even for q=5, where the correlation length of the system amount to several thousands of the lattice spacing and other methods such as the Monte Carlo simulations cannot give any meaningful result.Secondly we calculated the low-temperature expansion series for the three kinds of quantities concerning the interface width for the Ising model in three dimensions and for the Absolute value SOS model and Discrete Gaussian model on the square lattice and on the triangular lattice, respectively, to 17th, 23rd, 22nd, 24th and 22nd order in the expansion parameter for the respective model. Assuming the critical exponent of the each quantity that is predicted by the renormalization group arguments, we obtained from the series the value of the roughening transition point for each model. They agree quite well with the results of the Monte Carlo simulation for the former three models and they give new prediction for the latter two models.