Decay of Order in Isotropic Systems of Restricted Dimensionality. II. Spin Systems

Abstract

The ordering of one- and two-dimensional spin systems of finite thickness and cross section is considered in the presence and absence of a symmetry-breaking magnetic field. The exchange interactions are allowed to vary randomly or regularly throughout the lattice. It is shown rigorously by applying Bogoliubov's inequality to a subdomain of the system that, provided the (suitably averaged) exchange interactions do not fall off too slowly, no spontaneous ordering can occur. Explicit bounds on the spin-spin correlation function, summed over the sites in a subdomain, are obtained which indicate how the short-range order decays with distance. Detailed numerical plots for the order as a function of the subdomain size are presented for various realistic values of the temperature. Conditions under which these curves yield bounds on the spatial decay of the spin-spin correlation function are also discussed.