Critical and reentrant phenomena in the Blume–Emery–Griffiths model with attractive biquadratic interaction

Using the mean field approximation based on the Bogoliubov inequality for free energy, we investigate the effects of the biquadratic exchange interaction on the critical behavior and phase diagram of the mixed spin (1,2) Blume–Emery–Griffiths model. We first focus on a specific case of the model: when the interaction parameter $$K=0$$ K = 0 , corresponding to the Blume–Capel model. For the attractive Blume–Emery–Griffiths model, we present the phase diagram in the temperature-crystal field plane. The phase diagram is significantly influenced by the value of the biquadratic exchange interaction K. For small values of K, similar to the Blume–Capel model, the phase diagram exhibits first and second order phase transition lines separating ordered and disordered phases, with a tricritical point marking the boundary between these regions. As K increases, the phase diagram changes significantly, with the appearance of reentrant and double reentrant phenomena. Graphic abstract The presence of a biquadratic interaction on mixed spin-1 and spin-2 Blume-Capel model gives rise to some important phenomena; The system present different phase transitions, namely, first- and second-order and reentrant phenomena; The effect of single-ion anisotropy and biquadratic interaction on magnetizations is established; A discussion with other approximation results is carried out.