Generalized spherical version of the Blume-Emery-Griffiths model with ferromagnetic and antiferromagnetic interactions

We have investigated analytically the phase diagram of a generalized spherical version of the Blume-Emery-Griffiths model that includes ferromagnetic or antiferromagnetic spin interactions as well as quadrupole interactions in zero and nonzero magnetic field. We show that in three dimensions and zero magnetic field a regular paramagnetic-ferromagnetic (PM-FM) or a paramagnetic-antiferromagnetic (PM-AFM) phase transition occurs whenever the magnetic spin interactions dominate over the quadrupole interactions. However, when spin and quadrupole interactions are important, there appears a reentrant FM-PM or AFM-PM phase transition at low temperatures, in addition to the regular PM-FM or PM-AFM phase transitions. On the other hand, in a nonzero homogeneous external magnetic field H, we find no evidence of a transition to the state with spontaneous magnetization for FM interactions in three dimensions. Nonetheless, for AFM interactions we do get a scenario similar to that described above for zero external magnetic field, except that the critical temperatures are now functions of H. We also find two critical field values, H_c1, at which the reentrance phenomenon disappears and H_c2 (H_c1~0.5H_c2), above which the PM-AFM transition temperature vanishes.