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

Abstract

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 $J_1$ and $J_2$ for the first and the second layer, respectively, and interlayer coupling constant $J_3$ 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 $|J_3\overrightarrow{|}0,$ $\Delta T_c\equiv \left[T_c{(J}_3\right)-T_c\mathrm{}\left(0\right)\mathrm{}\left]{/T}_c\mathrm{}\right(0\left)\mathrm{}\sim |J_3\right]{/J}_1{|}^{1/\psi }$, where $T_c{(J}_3)$ is the phase-transition temperature for the system with interlayer coupling constant $J_3$ and the shift exponent $\psi$ is 1 for $J_1{=J}_2$ and is 0.5 for $J_1\ne J_2$. Such results are consistent with predictions of a scaling theory. We also derive equations for $\Delta T_c$ when $|J_3|$ approaches $\infty$.