Phase Diagram of the Blume-Emery-Griffiths-Vannimenus Model on the Cayley Tree

We study the phase diagram of the Blume-Emery-Griffiths-Vannimenus model on a Cayley tree with competing nearest-neighbour couplings and next-nearest-neighbour couplings and show that a detailed study of its properties can be carried out with essentially exact results, using rather simple numerical methods. In addition to the expected paramagnetic and ferromagnetic phases, we find modulated phase for Blume-Emery-Griffiths model and hence for Blume-Emery-Griffiths-Vannimenus model also.