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

Abstract: For the repulsive Blume-Emery-Griffiths model the phase diagram in the space of three fields, temperature (T), crystal field ($\Delta$), and magnetic field (H), is computed on a complete graph, in the canonical and microcanonical ensembles. For weak strength of the biquadratic interaction (K), there exists a tricritical point in the phase diagram where three critical lines meet. As K decreases below a threshold value(which is ensemble dependent), new multicritical points like the critical end point and bicritical end point arise in the (T,$\Delta$) 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. Exactly at 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 K.