Analytical results for a continuum model of crystalline tensionless surfaces. I. Variational mean field study

We study analytically the equilibrium and near-equilibrium properties of a model of surfaces relaxing via linear surface diffusion and subject to a lattice potential. We employ the variational mean field formalism introduced by Saito for the study of the sine Gordon model. In equilibrium, our variational theory predicts a first order roughening transition between a flat low temperature phase and a rough high temperature phase with the properties of the linear molecular beam epitaxy equation. The study of a Gaussian approximation to the Langevin dynamics of the system indicates that the surface shows hysteresis when we continuously tune temperature. Out of equilibrium, this Langevin dynamics approach shows that the surface mobility can have different behaviours as a function of a driving flux. Some considerations are made regarding different dimensionalities and underlying lattices, and connections are drawn to related models or different approaches to the same model we study.