Multi-component spin model on a Cayley tree

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Abstract
A general spin model on the Cayley tree lattice, which includes both the q component Potts (1952) and the Ashkin-Teller (1943) models, is considered. The free energy in zero field is evaluated in a closed form and found to be analytic in temperature. The model exhibits no long-range order in the sense that the probability of finding two sites far away to be in spin states alpha and beta is constant, independent of alpha and beta . The susceptibility per site, chi R, for a region R in the centre of the lattice, defined to be the summation of the site-site correlations between R and the whole lattice L. For the linear size of R to be any finite fraction of that of L, chi R diverges at the Bethe-Peierls temperature(s) TBP, while for R identical to L, chi R diverges at temperature(s) different from TBP.