The random field Ising model with an asymmetric trimodal probability distribution

TL;DR

This paper investigates the phase transitions of the random field Ising model with an asymmetric trimodal distribution using mean field theory, revealing second and first order transitions, tricritical points, and reentrant behavior.

Contribution

It introduces an extension of the random field distribution to include non-magnetic sites, analyzing its effects on phase transitions and tricritical points within the mean field approximation.

Findings

Existence of second order phase transitions.

Presence of first order transitions and tricritical points.

Reentrant phase behavior observed for certain parameters.

Abstract

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 $P(h_{i})=p \delta(h_{i}-h_{0}) + q \delta (h_{i}+h_{0}) + r \delta(h_{i})$, where the 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 and $h_{0}$ 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 $p, q$ and $h_{0}$, 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 $h_{0}$.