Bethe-Peierls approximation with competing order parameters

We study an alloy of Ising spins between a ferromagnetic and an antiferromagnetic species. Pairs of $\mathrm{AA}$, $\mathrm{AB}$, and $\mathrm{BB}$ neighbors are characterized by different coupling constants $J_{\mathrm{AA}}$, $J_{\mathrm{AB}}$, and $J_{\mathrm{BB}}$. Our approach is based on a generalization of the Bethe-Peierls approximation to the case of compositional disorder. We find a phase diagram with a bicritical or tetracritical point depending on the strength of the coupling $J_{\mathrm{AB}}$ between species.