Lower Critical Dimension of Metallic Vector Spin-Glasses

We argue that when a short-range spin-glass system is below its lower critical dimension ${d}_{l}$, which seems to be the case for isotropic vector spins in three dimensions, then the corresponding Ruderman-Kittel-Kasuya-Yosida (RKKY) system is in a different universality class and at its lower critical dimension. For dimensions greater than ${d}_{l}$, the RKKY and short-range systems have the same critical behavior. This appears to apply to Ising spins, and to anisotropic vector-spin models for which we discuss the dependence of ${T}_{c}$ on anisotropy.