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

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-g Ising model includes the spin-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 xi(T,h) for the simple Ising model and BEG model, which show interesting scaling and divergent behavior as 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.