Phase diagrams and multicritical points in randomly mixed magnets. I. Mixed anisotropies

The critical behavior of a quenched random alloy of two materials with competing anisotropies is studied by renormalization-group techniques. Averaging over the random variables yields a translationally invariant effective Hamiltonian, in which the $m$-component spin is replaced by a $\mathrm{nm}$-component spin vector and the limit $n\to 0$ is taken at the end of the calculation. The only physical stable fixed point is "decoupled," leading to a Hamiltonian which separates into two parts, each depending on $m_1$ or $m_2=m-m_1$ spin components only, asymptotically close to criticality. This leads to breakdown of standard scaling in the vicinity of the multicritical point. The resulting phase diagram exhibits two critical lines, corresponding to ordering of only $m_1$ (or $m_2$) spin components, intersecting at an angle at a tetracritical point. The phases where only $m_1$ or $m_2$ components order are separated by an intermediate phase, where all components order. The effect of corrections to scaling on the shape of the critical lines in the vicinity of the tetracritical point is examined. Experiments on rare-earth alloys and layered materials are discussed.