Variational mean-field study of a continuum model of crystalline tensionless surfaces

We study analytically the equilibrium and near-equilibrium properties of a model of a d-dimensional surface 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. Moreover, the study of a Gaussian approximation to the Langevin dynamics of the system indicates that the surface shows hysteresis when temperature is continuously tuned. Out of equilibrium, these approximate Langevin dynamics show that the surface mobility can have different behaviors as a function of a driving flux. Some considerations are made regarding different underlying lattices, and connections are drawn to related models or different approaches to the same model we study.