±J model on the Bethe lattice with crystal field interaction

• ±J±J model was studied on the Bethe lattice for the spin-1 Blume–Capel model.
• Besides the first- and second-order phase transitions, spin glass phase was also found.
• Tricritical and multicritical points were found.
• The coordination number dependencies were investigated.
• Spin glass (SG) phase was observed without needing SG order parameter.

±J   model was exploited on the Bethe lattice for the spin-1 Blume–Capel model. Either ferromagnetic (J>0J>0) or antiferromagnetic (J<0J<0) interactions are assumed between the nearest-neighbor (NN) spins on the Bethe lattice for given probabilities p or 1−p, respectively. The phase diagrams of the model were calculated exactly on the temperature (T)–probability (p) or crystal field (D) planes for given coordination numbers q=3, 4 and 6. In addition to the first- and second-order phase transitions, another kind phase transition with zero magnetization and quadrupolar order-parameter presenting a jump was obtained above the second-order phase transition temperature which seems to be indicating the spin glass phase. When the lines of these three phases combine is a multicritical point and the first two combine is a tricritical point. In the (T,p) planes, the lower critical dimensions (the value of q) of obtaining any phase transition increase with increasing negative D values.