Phase diagrams of the Blume-Capel model on the two-fold Cayley tree

In this study, phase diagrams were obtained by examining the magnetization process and phase transitions of the spin-1 Blume-Capel model on a two-layer Cayley tree in the presence of an external magnetic field. The model Hamiltonian includes the nearest neighbor interaction between spin-1 atoms, the single-ion anisotropy parameter, and the external magnetic field. First, using the exact recursion method, the exact recursion relations on the two-fold Cayley tree in the presence of an external magnetic field were obtained after lengthy calculations. Numerical calculations provided a detailed examination of the sublattice and total magnetizations for the antiferromagnetic interaction. The possibility of obtaining re-entrant phase transition behavior in the presence of external field in finite temperature phase diagrams was also investigated. It has been shown that it exhibits re-entrant behavior for sufficiently high lattice coordination numbers (q=6, 8 and 12), close to the ground state phase boundary between antiferromagnetic (AFP) and disordered phases (DP), only in the limited range of the interaction parameter range that defines the model. Thus, the antiferromagnetic spin-1 BC model described in the two-fold Cayley tree shows basically the same critical behavior, even though it differs from each other for equal coordination numbers, although there are different geometries between both lattices (Bethe lattice and two-fold Cayley tree). On the other hand, for a two-fold Cayley tree with coordination numbers q=3, 4 and 5, there is no re-entrant behavior at the boundaries of the finite-temperature phase diagrams. Temperature dependencies of both sublattice magnetizations were also investigated and discussed in detail.