The random field Ising model with an asymmetric and anisotropic bimodal 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 asymmetric and anisotropic bimodal probability distribution P ( h i ) = p δ ( h i − h 0 ) + q δ ( h i + λ ∗ h 0 ) , where the site probabilities p , q take on a value within the interval [ 0 , 1 ] with the constraint p + q = 1 , h i is the random field variable with strength h 0 and λ is the competition parameter, which is the ratio of the strength of the random magnetic field in the two directions + z and − z ; λ is considered to be positive, resulting in competing random fields. For small and large values of p ( p < 13 − 1 3 26 or p > 13 + 1 3 26 , respectively) the phase transitions are exclusively of second order, but for 13 − 1 3 26 ≤ p ≤ 13 + 1 3 26 they are of second order for high temperatures and small random fields and of first order for small temperatures and high/small random fields irrespective of the λ -value; in the latter case the two branches are joined smoothly by a tricritical point confirming, in this way, the existence of such a point. In addition, re-entrant phenomena can be seen for appropriate ranges of the temperature and random field for a specific p -value. Using the variational principle, we determine the equilibrium equation for the magnetization, and solve it for both transitions and at the tricritical point, in order to determine the magnetization profile with respect to h 0 .