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天津大学学报：英文版

用 Ising 模型计算磁性薄膜的临界指数β(L)(英文)

霍炳海, 区镜添

Published Jan 1, 1998• 0 citations
OpenAlex

Abstract

近年来出现的变分累积展开法（ＶＣＥ）已被应用于统计物理中伊辛（Ｉｓｉｎｇ）模型的研究．本文利用ＶＣＥ方法，计算了Ｌ层磁性薄膜的临界温度Ｔｃ（Ｌ）和临界指数β（Ｌ）．当磁性薄膜的层数Ｌ＞１时，本文的理论计算结果与实验结果大体相符．对于Ｌ＝１（二维）这一特例，本文计算结果也与有关报道相同．

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近年来出现的变分累积展开法（ＶＣＥ）已被应用于统计物理中伊辛（Ｉｓｉｎｇ）模型的研究．本文利用ＶＣＥ方法，计算了Ｌ层磁性薄膜的临界温度Ｔｃ（Ｌ）和临界指数β（Ｌ）．当磁性薄膜的层数Ｌ＞１时，本文的理论计算结果与实验结果大体相符．对于Ｌ＝１（二维）这一特例，本文计算结果也与有关报道相同．
在局域密度近似的框架内(其中气体在每一点都可以看成是局域均匀的),得到谐振势中相互作用费米气体的能量密度.在这些表达式的基础上,研究了零温时约束气体的热力学稳定性条件,给出了一维、二维、三维系统稳定条件的解析表达式.此外,绘出了不同维度系统的稳定性相图.结果表明:在本研究中的所有维度中,在特定粒子数密度的条件下,排斥相互作用或无相互作用系统是稳定的.然而,对于吸引相互作用系统,只有在相互作用很弱的条件下,系统才是稳定的.对于一维和三维系统来说,相互作用强度的临界值与粒子数密度紧密相关,而对于二维系统而言,相应的临界值同粒子数密度不存在明显关联性.
The variational-cumulant-expansion (VCE) method is adopted to calculate the Heisenberg model. The free energy of the model is expanded to the third order, and the analytical formulas for the critical temperature Tc{\mathit{T}}_{\mathit{c}}Tc​ are given for each order. The first order of VCE corresponds to the mean-field theory, the second order gives the same result for Tc{\mathit{T}}_{\mathit{c}}Tc​ as the 1/z expansion, and the third order gives a more complicated correction. As a comparison, the results of the two-dimensional (2D) Ising model for VCE to the seventh order are shown. We carry out a trial to predict the critical temperature of the infinite order of the VCE, and get a very accurate result for the 2D Ising model. A prediction of Tc{\mathit{T}}_{\mathit{c}}Tc​ for the Heisenberg model is also made.
It is shown for the Ising model that the linked-cluster expansion (LCE) is a special case of the variational cumulant expansion (VCE) in the high-temperature limit. The VCE was originally developed and formulated to treat lattice-gauge-field models. The free energy calculated from the VCE with a trial Hamiltonian of mean-field form in the high-temperature limit is in fact identical to that from the LCE. Thus, a more general expansion that is valid for the whole temperature range is established for the treatment of Ising systems.