Surfaces and interfaces of lattice models: mean-field theory as an area-preserving map

Abstract

Mean-field theory for one-dimensionally inhomogeneous magnetic systems is formulated as an area-preserving map. The map and its associated boundary conditions are derived for nearest-neighbor Ising interactions. The corresponding continuum theory is also constructed. These mappings are two dimensional. Their phase portraits are exhibited and applied to the study of a representative set of surface and interface phenomena, including interfacial structure, surface phase transitions, wetting, prewetting, and layering. The methods developed lend themselves to easy and physical visualization of the types of solutions which the mean-field theory can have, even in rather complex situations. They also make explicit the fundamental differences between continuum mean-field theory (which is integrable) and discrete mean-field theory (which is not).