Critical Temperatures of Anisotropic Ising Lattices. II. General Upper Bounds

Abstract

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 $\frac{k{T}_c}{2dJ}\le 1-\left(\frac{1}{2d}\right)-\left(\frac{1}{3d^2}\right)+O\left(\frac{1}{d^3}\right)$ as $d\to \infty$. For anisotropic hypercubical lattices with interactions $J_i$ ($i=1, \cdots d$), the equality $\frac{k{T}_c}{J_1}=2\mathrm{}{[\ln {\eta }^{-1\mathrm{}}-\mathrm{lnln}{\eta }^{-1\mathrm{}}+O(1\left)\right]}^{-1\mathrm{}}$ is proved for all $d\ge 2$ in the limit $\eta =\frac{(J_2+J_3+\cdots +J_d)}{J_1}\to 0$.