Highand Lowtemperature expansion for the spin statistical systems using the finite lattice method
Project/Area Number  08640494 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
物性一般(含基礎論)

Research Institution  Osaka Prefectural College of Technology 
Principal Investigator 
ARISUE Hiroaki Osaka Prefectural College of Technology, Professor, 助教授 (10175987)

Project Fiscal Year 
1996 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥2,000,000 (Direct Cost : ¥2,000,000)
Fiscal Year 1998 : ¥200,000 (Direct Cost : ¥200,000)
Fiscal Year 1997 : ¥200,000 (Direct Cost : ¥200,000)
Fiscal Year 1996 : ¥1,600,000 (Direct Cost : ¥1,600,000)

Keywords  Ising model / Potts model / hightemperature expansion / finite lattice method / phase transition / critical exponent / magnetic susceptibility / correlation length / イジング模型 / ポッツ模型 / 高温展開 / 有限格子法 / 相転移 / 臨界指数 / 帯磁率 / 相関距離 / 低温展開 / Potts模型 / largeq展開 / Patts模型 / SOS模型 / 比熱 
Research Abstract 
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 lowtemperature expansion series of the free energy, the magnetization and the magnetic susceptibility in the qstate Potts model in two dimensions for all of q=550, 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 largeq 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 lowtemperature 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.

Report
(5results)
Research Output
(18results)