Ground State, Spectrum, and Phase Diagram of Two-Dimensional Incommensurate Crystals

A model of a monolayer of atoms adsorbed on a crystalline surface is discussed. When the initial incommensurability, $\delta$, between a reciprocal lattice vector of the monolayer and one of the substrate is small, the monolayer has a one-dimensional superstructure with period depending on $\delta$. For $\delta >{\delta }_c\mathrm{}\left(T\right)\mathrm{}$ the state is incommensurate; ${\delta }_c\mathrm{}\left(T\right)\mathrm{}$ decreases and vanishes at a $T_c$ given by the elastic moduli of the monolayer. The spectrum of small oscillations has two acoustic branches, one vanishing at the phase transition.