Absence of first order transition in the random crystal field Blume-Capel model on a fully connected graph

In this paper we solve the Blume-Capel model on a complete graph in the presence of random crystal field with a distribution, $P(Δ_i) =pδ(Δ_i-Δ)+(1-p) δ(Δ_i+Δ)$, using large deviation techniques. We find that the first order transition of the pure system is destroyed for $0.046<p<0.954$ for all values of the crystal field, $Δ$. The system has a line of continuous transition for this range of $p$ from $-\infty <Δ<\infty$. For values of $p$ outside this interval, the phase diagram of the system is similar to the pure model, with a tricritical point separating the line of first order and continuous transitions. We find that in this regime, the order vanishes for large $Δ$ for $p<0.046$(and for large $-Δ$ for $p>0.954$) even at zero temperature.

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