Relation between lattice and continuum theories of two-dimensional solids

Abstract

A general model is proposed to describe the melting of adsorbed monolayers with a lattice constant which is commensurate to the underlying substrate. The model introduced here is a vector generalization of the discrete planar (clock) model. Dislocations are incorporated in the theory, and the translational invariance of the substrate is preserved. Exact duality relations can be used to obtain properties of the model. The phases which occur and which are considered in most detail can be described by all combinations of long-range, quasi-long-range (algebraic), and short-range order in the two perpendicular displacement fields. Depending on the symmetry of the ordered phase relative to the substrate, an intermediate "floating" phase, with algebraic decay of positional order can exist on a triangular lattice if the ratio of lattice constants of the overlayer to the substrate exceeds $2\sqrt{3}$.