The exact phase diagrams of spin-1 Ising model on a two-layer Bethe lattice

The two-layer spin-1 Ising model on the Bethe lattice is studied in terms of the intralayer coupling constants J1 and J2 of the two layers, interlayer coupling constant J3 between the layers and the external magnetic fields, which are coupled to the two layers and assumed to be different for each layer, for given values of the coordination number q by using the recursion relation scheme. The ground-state configurations of the system are obtained on the (J2/|J1|,J3/q|J1|) planes depending on J1<0 or J1>0. Then, the phase diagram of the system is obtained on the (kT/J1,J3/J1) plane for given values of α=J2/J1 and q in zero external magnetic fields. It was found that the system presents both first- and second-order phase transitions for all values of q. Besides, we also present the thermal change of the total and staggered magnetizations of the two layers and also the spin-spin correlation function between the nearest-neighbor spins of the adjacent layers.