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

TL;DR

This paper investigates the phase transitions and critical phenomena of the random field Ising model with an asymmetric, anisotropic trimodal distribution, revealing complex behaviors including first and second order transitions, tricritical points, and re-entrant phenomena.

Contribution

It introduces a novel asymmetric, anisotropic trimodal probability distribution for the random field and analyzes its effects on phase transitions in the Ising model, including tricritical points and re-entrant behavior.

Findings

Mainly second order phase transitions observed

Presence of tricritical points and re-entrant phenomena

Complex phase diagram with multiple transition types

Abstract

The Ising model in the presence of a random field, drawn from the asymmetric and anisotropic trimodal probability distribution $P(h_{i})=p\; \delta(h_{i}-h_{0}) + q \delta (h_{i}+ \lambda *h_{0}) + r \delta (h_{i})$, 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); $\lambda$ 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 $\lambda$, $p$ 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.