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The spin-1/2 Ising model on a square lattice, with fluctuating bondinteractions between nearest neighbors and in the presence of a random magneticfield, is investigated within the framework of the effective field theory basedon the use of the differential operator relation. The random field is drawnfrom the asymmetric and anisotropic bimodal probability distribution$P(h_{i})=p \delta(h_{i}-h_{1}) + q \delta (h_{i}+ ch_{1})$, where the siteprobabilities $p,q$ take on values within the interval $[0,1]$ with theconstraint $p+q=1$; $h_{i}$ is the random field variable with strength $h_{1}$and $c$ the competition parameter, which is the ratio of the strength of therandom magnetic field in the two principal directions $+z$ and $-z$; $c$ isconsidered to be positive resulting in competing random fields. The fluctuatingbond is drawn from the symmetric but anisotropic bimodal probabilitydistribution $P(J_{ij})=\frac{1}{2}\{\delta(J_{ij}-(J+\Delta)) + \delta(J_{ij}-(J-\Delta))\}$, where $J$ and $\Delta$ represent the average value andstandard deviation of $J_{ij}$, respectively. We estimate the transitiontemperatures, phase diagrams (for various values of system's parameters$c,p,h_{1},\Delta$), susceptibility, equilibrium equation for magnetization,which is solved in order to determine the magnetization profile with respect to$T$ and $h_{1}$.

physica a statistical mechanics and its applications

Effective field theory for the Ising model with a fluctuating exchange  integral in an asymmetric bimodal random magnetic field: A differential  operator technique