Phase diagram of the repulsive Blume-Emery-Griffiths model in the presence of an external magnetic field on a complete graph

Using the repulsive Blume-Emery-Griffiths model, we compute the phase diagram in three field spaces, temperature (T), crystal field (Δ), and magnetic field (H) on a complete graph in the canonical and microcanonical ensembles. For low biquadratic interaction strengths (K), a tricritical point exists in the phase diagram where three critical lines meet. As K decreases below a threshold value (which is ensemble dependent), new multicritical points such as the critical end point and the bicritical end point arise in the (T, Δ) plane. For K > -1, we observe that the two critical lines in the H plane and the multicritical points are different in the two ensembles. At K = -1, the two critical lines in the H plane disappear, and as K decreases further, there is no phase transition in the H plane. At exactly K = -1, the two ensembles become equivalent. Beyond that, for all K < -1, there are no multicritical points, and there is no ensemble inequivalence in the phase diagram. We also study the transition lines in the H plane for positive K, i.e. for attractive biquadratic interaction. We find that the transition lines in the H plane are not monotonic in temperature for large positive values of K.