The random field Ising model with an asymmetric trimodal probability distribution

The Ising model in the presence of a random field is investigated within the mean field approximation based on Landau expansion. The random field is drawn from the trimodal probability distribution P(hi)=pδ(hi−h0)+qδ(hi+h0)+rδ(hi)P(hi)=pδ(hi−h0)+qδ(hi+h0)+rδ(hi), where the probabilities p,q,rp,q,r take on values within the interval [0,1][0,1] consistent with the constraint p+q+r=1p+q+r=1 (asymmetric distribution), hihi is the random field variable and h0h0 the respective strength. This probability distribution is an extension of the bimodal one allowing for the existence in the lattice of non magnetic particles or vacant sites. The current random field Ising system displays second order phase transitions, which, for some values of p,qp,q and h0h0, are followed by first order phase transitions, thus confirming the existence of a tricritical point and in some cases two tricritical points. Also, reentrance can be seen for appropriate ranges of the aforementioned variables. Using the variational principle, we determine the equilibrium equation for magnetization, solve it for both transitions and at the tricritical point in order to determine the magnetization profile with respect to h0h0.

► Existence of Tricritical Points for specific values of partial probabilities.
► Appearance of reentrance.
► Closed loop magnetization profiles with higher order critical points.
► Residual magnetization in the system.