Critical behavior of random systems

The critical behavior of magnetic systems with nonmagnetic impurities is analyzed. It is argued that most magnetic systems should be described by the random-uniaxial-anisotropy model (RAM), rather than the random-exchange model. The crossover exponent ${\psi }_m$ associated with random uniaxial anisotropy in $m$-component systems satisfies ${\psi }_m=2\mathrm{}{\phi }_m-2+{\alpha }_m\sim 0.3>0$ for $m\gtrsim 2$ where ${\phi }_m$ is the anisotropy crossover exponent and ${\alpha }_m$ is the specific-heat exponent of the pure system. The critical behavior of these systems is therefore expected to be different from that of the pure ones. The critical behavior of the RAM with cubic- and higher-order nonrandom anisotropy terms (which are always present in models appropriate for nonamorphous compounds) is studied using renormalization-group calculations in $d=4-\epsilon$ dimensions. It is also argued that the multicritical behavior of randomly mixed magnets with competing order parameters is not determined by the decoupled fixed point, as suggested in previous studies.