Theory of Toeplitz Determinants and the Spin Correlations of the Two-Dimensional Ising Model. III

Abstract

We study the asymptotic behavior, for large separations, of the spin-spin correlation function $〈{\sigma }_{0,0}{\sigma }_{M,N}〉$ in the two-dimensional Ising model, where the two spins are not necessarily on the same row. Besides the limiting value for infinite separation, which is the square of the spontaneous magnetization, we evaluate the two leading terms in the asymptotic expression in each of the two cases $T<\mathrm{}{T}_c$ and $T>\mathrm{}{T}_c$. It is found that the nearest singularity of the generating function for the correlation is quite simple in the case $T>\mathrm{}{T}_c$, but much more complicated for $T<\mathrm{}{T}_c$. In an Appendix, we also give exactly in a very simple form the correlation $〈{\sigma }_{0,0}{\sigma }_{N,N}〉$ for symmetrical Ising lattice at the critical temperature $T_c$.