Exact phase diagrams for an Ising model on a two-layer Bethe lattice

Using an iteration technique, we obtain exact expressions for the free energy and the magnetization of an Ising model on a two-layer Bethe lattice with intralayer coupling constants J1 and J2 for the first and the second layer, respectively, and interlayer coupling constant J3 between the two layers; the Ising spins also couple with external magnetic fields, which are different in the two layers. We obtain exact phase diagrams for the system and find that when /J3/-->0, DeltaT(c) identical with[T(c)(J3)-T(c)(0)]/T(c)(0) approximately [J3]/J(1)/(1/psi), where T(c)(J3) is the phase-transition temperature for the system with interlayer coupling constant J3 and the shift exponent psi is 1 for J(1)=J(2) and is 0.5 for J1 not equal to J2. Such results are consistent with predictions of a scaling theory. We also derive equations for DeltaT(c) when /J3/ approaches infinity.