The solid-on-solid (SOS) model of a crystal surface. Below the roughening temperature TR the surface is smooth (faceted). Above TR it roughens — the height-height correlation diverges logarithmically.
Crystal surface height profile (3D view)
Height-height correlation ⟨[h(r)−h(0)]²⟩ vs log(r)
Parameters
W² = 0
SOS Hamiltonian: H = J Σ|h_i − h_j|
Smooth phase: ⟨h²⟩ finite Rough phase: ⟨h²⟩ ~ log L
T_R ≈ J/(k_B π²/18) (approx) Related to BKT transition