Effect of random field disorder on the first order transition in $p$-spin interaction model

We study the random field $p$-spin model with Ising spins on a fully connected graph using the theory of large deviations in this paper. This is a good model to study the effect of quenched random field on systems which have a sharp first order transition in the pure state. For $p=2$, the phase-diagram of the model, for bimodal distribution of the random field, has been well studied and is known to undergo a continuous transition for lower values of the random field ($h$) and a first order transition beyond a threshold, $h_{tp}(\approx 0.439)$. We find the phase diagram of the model, for all $p \ge 2$, with bimodal random field distribution, using large deviation techniques. We also look at the fluctuations in the system by calculating the magnetic susceptibility. For $p=2$, beyond the tri-critical point in the regime of first order transition, we find that for $h_{tp}<h<0.447$, magnetic susceptibility increases rapidly (even though it never diverges) as one approaches the transition from the high temperature side. On the other hand, for $0.447<h \le 0.5$, the high temperature behaviour is well described by the Curie-Weiss law. For all $p \ge 2$, we find that for larger magnitudes of the random field ($h>h_o=1/p!$), the system does not show ferromagnetic order even at zero temperature. We find that the magnetic susceptibility for $p \ge 3$ is discontinuous at the transition point for $h<h_o$.