Phase Diagram of the Two-Dimensional Ising Model with Random Competing Interactions

An Ising model with ferromagnetic nearest-neighbor interactions $J_{1}$ ($J_{1}>0$) and random next-nearest-neighbor interactions [$+J_{2}$ with probability $p$ and $-J_{2}$ with probability $(1-p)$; $J_{2}>0$] is studied within the framework of an effective-field theory based on the differential-operator technique. The order parameters are calculated, considering finite clusters with $n=1,2, {\rm and} 4$ spins, using the standard approximation of neglecting correlations. A phase diagram is obtained in the plane temperature versus $p$, for the particular case $J_{1}=J_{2}$, showing both superantiferromagnetic (low $p$) and ferromagnetic (higher values of $p$) orderings at low temperatures.