The random field Ising model with an asymmetric and anisotropic trimodal probability distribution

Abstract

The Ising model in the presence of a random field, drawn from the asymmetric and anisotropic trimodal probability distribution $P(h_{i})=p\; δ(h_{i}-h_{0}) + q δ(h_{i}+ λ*h_{0}) + r δ(h_{i})$, is investigated. The partial probabilities $p, q, r$ take on values within the interval $[0,1]$ consistent with the constraint $p+q+r=1$, asymmetric distribution, $h_{i}$ is the random field variable with basic absolute value $h_{0}$ (strength); $λ$ is the competition parameter, which is the ratio between the respective strength of the random magnetic field in the two principal directions $(+z)$ and $(-z)$ and is positive so that the random fields are competing, anisotropic distribution. 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 mainly second order phase transitions, which, for some values of $p, q$ and $h_{0}$, are followed by first order phase transitions joined smoothly by a tricritical point; occasionally, two tricritical points appear implying another second order phase transition. In addition to these points, re-entrant phenomena can be seen for appropriate ranges of the temperature and random field for specific values of $λ$, $p$ and $q$. Using the variational principle, we write down the equilibrium equation for the magnetization and solve it for both phase transitions and at the tricritical point in order to determine the magnetization profile with respect to $h_{0}$, considered as an independent variable in addition to the temperature.