  We apply the Nozieres-Gallet dynamic renormalization group (RG) scheme to a
continuum equilibrium model of a d-dimensional surface relaxing by linear
surface tension and linear surface diffusion, and which is subject to a lattice
potential favoring discrete values of the height variable. The model thus
interpolates between the overdamped sine-Gordon model and a related continuum
model of crystalline tensionless surfaces. The RG flow predicts the existence
of an equilibrium roughening transition only for d = 2 dimensional surfaces,
between a flat low-temperature phase and a rough high-temperature phase in the
Edwards-Wilkinson (EW) universality class. The surface is always in the flat
phase for any other substrate dimensions d > 2. For any value of d, the linear
surface diffusion mechanism is an irrelevant perturbation of the linear surface
tension mechanism, but may induce long crossovers within which the scaling
properties of the linear molecular-beam epitaxy equation are observed, thus
increasing the value of the sine-Gordon roughening temperature. This phenomenon
originates in the non-linear lattice potential, and is seen to occur even in
the absence of a bare surface tension term. An important consequence of this is
that a crystalline tensionless surface is asymptotically described at high
temperatures by the EW universality class.
