Fractal Configurations of the Two- and Three-Dimensional Ising Models at the Critical Point
Fractal Configurations of the Two- and Three-Dimensional Ising Models at the Critical Point
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二维和三维 Ising 模型在临界点的分形结构
DOI：
10.1143/ptp.77.1391
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发表时间：
1987
影响因子：
--
通讯作者：
Masuo Suzuki
中科院分区：
文献类型：
--
作者：
N. Ito;Masuo Suzuki
本文研究了Ising模型在临界点T_c处的分形结构。用蒙特卡罗方法数值计算了二维正方晶格的总磁化强度的分维D=1.86±0.01，三维简单立方晶格的分维D=2.46±0.01。这些值分别与D=d-fj/v=(d+r/v)/2的关系式D=d-fj/v=(d+r/v)/2，与由精确临界指数得到的D=1.875和由已知临界指数得到的D=2.48值符合得很好。在低于T_c的温度下观察到相应晶格的维度d，而在较高的温度下观察到随机渗流值d/2。
The fractal structure of the Ising model at the critical point Tc is studied in the present paper. The fractal dimensionality D of the total magnetization at Tc was estimated numerically as D = 1.86 ±O.Ol for the two-dimensional square lattice and D=2.46±O.Ol for the three-dimensional simple cubic lattice by Monte Carlo simulations. These values agree very well with the value D=1.875 obtained from the exact critical exponents and D=2.48 obtained from the known critical exponents, re ~pectively, through the relation D=d-fJ/v=(d+r/v)/2. This fractalness yields the hyperscaling relation dv=2fJ+r. It was also observed how the fractal nature of the relevant system disappears as the system deviates from the critical point. The dimensionality d of the relevant lattice is observed at temperatures lower than Tc and the random percolation value d/2 at higher temperatures.