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?