The spin-1 Blume-Capel model on the Bethe lattice in  ± J distribution with an adjustable parameter between FM and AFM phases

The spin-1 Blume-Capel (BC) model is studied on the Bethe lattice (BL) for the  ±  J distribution with a competing adjustable parameter α which alters the strength of bilinear exchange interaction parameter for the ferromagnetic phase (J > 0) with respect to antiferromagnetic phase (J < 0). The J > 0 and αJ < 0 values are also distributed throughout the BL with probabilities p and 1−p, respectively. The order-parameters are obtained on the BL in terms of exact recursion relations (ERR's) and their temperature (T) variations are studied to calculate the phase diagrams on the (α, T) planes for given values of p, crystal field (D) and coordination number q=3 corresponding to honeycomb lattice. It is found that the model gives both first- and second-order phase transitions and also tricritical points. In addition to the well known ordinary phases and TCP's, the spin glass phase and two more special points are also observed.