Solid-on-solid models

Question

I'm currently wondering about the so-called Chui-Weeks model 1 2, given by $$ H = J \sum_i |h_i - h_{i+1}| + K\sum_i \delta_{0,h_i}, $$ which is a type of solid-on-solid (SOS) model used to describe dynamics and properties of surfaces. Here $h_i \in [0,\infty)$ describes the height of the surface above some reference point, and the second term in the Hamiltonian is a so-called pinning-potential, which tries to keep the surface at height zero.

The system undergoes a Roughening transition at some temperature $T_R$, where the surface goes from being smooth (and effectively pinned to the zero level) to being rough and fluctuating to infinity.

I was wondering about generalizations of this model, where we keep the first interaction term but get rid of the pinning potential, while the height of the surface will no be restricted to only positive values anymore, i.e. $h_i \in (-\infty,\infty)$. Are there any arguments on if there's still a roughening transition happening at some finite temperature, and if so why that is? Or is there maybe already some literature on that model that I couldn't find?

Answer 1

First, I'd say that the case you describe corresponds to a wetting transition, rather than to a roughening transition: the system goes from a regime of complete wetting (the wall if fully covered by a mesoscopic layer) at small (negative) $K$ to a regime of partial wetting (the wall is covered by microscopic droplets) at large (negative) $K$.

Concerning the variant you are asking about:

• For a one-dimensional model without pinning potential, the interface will always be delocalized. It is very easy to prove, as this is simply the trajectory of a random walk on $\mathbb{Z}$ conditioned to be at $0$ at time $0$ and $N$.
• There is a roughening transition in dimension $2$ (that is, in the case of a function $h:\mathbb{Z}^2\to\mathbb{Z}$). This is a famous result first proved rigorously by Fröhlich and Spencer in this paper.
• In dimensions $d\geq 3$, there is no roughening transition, since the interface should be always localized (as can be guessed from the fact that even the rougher Gaussian free field has bounded variance when $d\geq 3$); this was proved in some cases, but I don't remember the references at the moment.

Of course, things are more interesting in the case where you have $h_i\in(-\infty,\infty)$ and a pinning potential. This case is very well understood and I can provide references if needed.