Critical Temperatures of Anisotropic Ising Lattices. II. General Upper Bounds
Critical Temperatures of Anisotropic Ising Lattices. II. General Upper Bounds
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各向异性伊辛晶格的临界温度。
DOI：
10.1103/physrev.162.480
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发表时间：
1967
期刊：
影响因子：
--
通讯作者：
M. Fisher
中科院分区：
文献类型：
--
作者：
M. Fisher
严格的上限的相关函数，并从那里的磁化率和磁化的自旋1/2伊辛模型与一般铁磁相互作用的生成函数的相应的晶格上的自避免随机行走。这些结果被用来表明，自发磁化消失，初始磁化率是有限的超过一定的温度T0，因此是一个边界的临界温度Tc。因此，证明了平均场和Bethe近似产生上界的Tc。给出了更强的界;具体地说，对于d维各向同性超立方格，当d→∞时，证明了kTc 2 dJ ≤ 1−（1 2 d）−（1 3 d2）+ O（1 d3）.对于具有相互作用Ji（i= 1，<$d）的各向异性超立方格，在极限η=（J2 + J3 +<$n + Jd）J1 → 0中，证明了对所有d≥ 2，等式kTc J1 = 2 [ln η− 1− lnln η− 1+ O（1）]− 1.
Rigorous upper bounds to the correlation functions and from there to the susceptibility and magnetization of spin-½ Ising models with general ferromagnetic interactions are obtained in terms of the generating functions for self-avoiding random walks on the corresponding lattice. These results are used to show that the spontaneous magnetization vanishes and the initial susceptibility is finite above a certain temperature T 0 which is thus a bound for the critical temperature T c. It is hence proved that the mean-field and Bethe approximations yield upper bounds for T c. Stronger bounds are presented; specifically, for d-dimensional isotropic hypercubical lattices, it is shown that k T c 2 d J≤ 1−(1 2 d)−(1 3 d 2)+ O (1 d 3) as d→∞. For anisotropic hypercubical lattices with interactions J i (i= 1,⋯ d), the equality k T c J 1= 2 [ln η− 1− lnln η− 1+ O (1)]− 1 is proved for all d≥ 2 in the limit η=(J 2+ J 3+⋯+ J d) J 1→ 0.