The spin- bilayer Bethe lattice with crystal field

The phase diagrams of the bilayer spin- Ising model on the Bethe lattice is analyzed by taking into account the intralayer coupling constants of the two layers (J1,J2), interlayer coupling constant between the layers (J3) and crystal field interaction (D) for the coordination number q=3 by using the exact recursion equations. First, the distinct ground state phase diagrams of the model are obtained on the (J2/J1,J3/qJ1) plane with J1>0 for given crystal field values. Then, the phase diagrams of the system are obtained on the (kT/J1,J3/J1) planes for given values of the crystal field and α=J2/J1 on the honeycomb lattice, i.e., q=3. It was found that the system presents both first- and second-order phase transitions, therefore, tricritical points, and splitting of the critical lines at negative values of the crystal field. The paramagnetic phases are also classified by studying the thermal variations of the quadrupolar moments.