Exact correlation functions of Bethe lattice spin models in external magnetic fields

Abstract

We develop a transfer matrix method to compute exactly the spin-spin correlation functions $〈{s}_{0}{s}_{n}〉$ of Bethe lattice spin models in the external magnetic field $h$ and for any temperature $T.$ We first compute $〈{s}_{0}{s}_{n}〉$ for the most general spin-$S$ Ising model, which contains all possible single-ion and nearest-neighbor pair interactions. This general spin-$S$ Ising model includes the spin-$\frac{1}{2}$ simple Ising model and the Blume-Emery-Griffiths (BEG) model as special cases. From the spin-spin correlation functions, we obtain functions of correlation length $\ensuremath{\xi}(T,h)$ for the simple Ising model and BEG model, which show interesting scaling and divergent behavior as $\stackrel{\ensuremath{\rightarrow}}{h}0$ and $T$ approaches the critical temperature ${T}_{c}.$ Our method to compute exact spin-spin correlation functions may be applied to other Ising-type models on Bethe and Bethe-like lattices.