The Blume-Emery-Griffiths Model with Competing Interactions on the Bethe Lattice

We study in detail the behaviour of a spin-1 model with competing interactions between next and next-nearest neighbours, biquadratic interactions between nearest neighbours, crystalline and external magnetic fields, on a Bethe lattice in the infinite coordination limit. We solve this model analytically and investigate its phase diagrams by numerically iterating the recursion relations on the Bethe lattice. The phase diagrams display more than one multiphase points and finite commensurate modulated phases both at finite and at zero temperature. The ground state of the system is also investigated. We observe rich phase diagrams with different ordered phases arranged in complex structures. Special attention is dedicated to the peculiar effects of biquadratic interactions and crystalline field, which in general distort and eventually suppress some of the commensurate ordered phases. The external field behaviour shows differences between 0 ordered phase and the paramagnetic phase for the first time in these models.