The random field Blume-Capel model on the Bethe lattice

The spin-1 Blume-Capel model is studied under the effect of a bimodal random magnetic field with equal probabilities on the Bethe lattice in terms of the exact recursion relations. The turning of magnetic field up or down with equal probability is the key of this work, since in this case one can observe the occurrence of phase transitions in contrary to the continuous magnetic field acting on the same model. Therefore, the phase diagrams are calculated on the crystal field-temperature planes for given values of the random magnetic field with the coordination numbers z = 3, 4 and 6. It is found that the model does give both second- and first-order phase transitions in addition to the tricritcal and another type of a critical point.