We obtain the phase diagram for the Blume-Capel model with bimodal distribution for random crystal fields, in the space of three fields: temperature(T ), crystal field(∆) and magnetic field (H). We find that three critical lines meet at a tricritical point, but only for weak disorder. As disorder strength increases there is no tricritical point in the phase diagram. We instead find a bicritical end point, where only two of the critical lines meet on a first order surface in the H = 0 plane. For intermediate strengths of disorder, the phase diagram has critical end points along with the bicritical end point. One needs to look at the phase diagram in the space of three fields to identify various such multicritical points.

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.