Spin-glass model with short- and long-range interactions

We describe a magnetic model with short-range couplings and an infinite-ranged Ising coupling whose strength is Gaussian distributed about zero. We show within the replica method that the model has a spin-glass transition temperature $T_g$ which always lies above the magnetic transition temperature $T_{c0\mathrm{}}$ of the short-range Hamiltonian taken alone, as long as the dimension $d<\mathrm{}4-2\mathrm{}\eta$ where the exponent $\eta$ describes the spin-spin correlation function for the short-range system. Two methods for computing the properties of the model are described, namely, systematic corrections to a uniform-field approximation and an expansion in the Edwards-Anderson order parameter $q$. Both give qualitatively similar results. We discuss evidence that the magnetization is always zero for all temperatures in this model. The model is qualitatively consistent with experiments on $\mathrm{K}{\mathrm{Mn}}_x{\mathrm{Zn}}_{1-x}{\mathrm{F}}_3$ and ${\mathrm{Eu}}_x{\mathrm{Sr}}_{1-x}\mathrm{S}$ if peaks which have been attributed to Bragg scattering are actually very narrow diffuse-scattering peaks, as recent experiments on ${\mathrm{Eu}}_x{\mathrm{Sr}}_{1-x}\mathrm{S}$ seem to suggest.