Phase diagrams and multicritical points in randomly mixed magnets. III. Competing spin-glass and magnetic ordering

The phase diagrams and critical properties of a quenched random alloy of a ferromagnet and an antiferromagnet (or of two antiferromagnets with different periodicities) are studied in the mean-field approximation and by renormalization-group techniques in $d=6-\epsilon$ dimensions, using the $n\to 0$ replica method. If only nearest-neighbor interactions are assumed, then one finds ferromagnetic ($F$), antiferromagnetic (AF), and spin-glass (SG) phases. As the average strength of the next-nearest-neighbor interactions is increased (e.g., by an additional component in the alloy), the $F$ and the AF phases approach each other, and one may find regions in parameter space where a mixed $F$-AF phase exists, as discussed in Paper II of this series. The phase diagram exhibits lines of multicritical points where the paramagnetic ($P$), $F$ (or AF) and SG phases coexist ($P$-SG-$F$ or $P$-SG-AF), or where the $P$, $F$, and AF phases coexist ($P-F$-AF). These lines meet at a new multicritical point, where all four phases coexist ($P$-SG-$F$-AF). Recent renormalization-group analysis of $P$-SG-$F$, by Chen and Lubensky, yields complex exponents for the $\mathrm{XY}$ and Heisenberg cases. We show that the phase diagram predicted by that analysis is difficult to understand even for the Ising case. The same method of analysis yields similar difficulties for $P$-SG-$F$-AF. A modified way to take the limit $n\to 0$, which resolves these difficulties, is presented. This modified way still lacks rigorous justification.