Exact Correlation Functions of Bethe Lattice Spin Models in External Fields

We develop a transfer matrix method to compute exactly the spin-spin correlation functions < s0sn > of Bethe lattice spin models in the external magnetic field h and for any temperature T . We first compute < s0sn > 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-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, ξ(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 Tc. Our method to compute exact spinspin correlation functions may be applied to other Ising-type models on Bethe and Bethe-like lattices.