Below T_R the crystal surface is smooth; above it becomes rough — a continuous phase transition of the interface itself
In the 2D Ising model with fixed +/− boundary conditions (top half ↑, bottom ↓), a domain wall forms. Below the roughening temperature T_R, the interface is localized and flat (roughness W ∝ log L, bounded). Above T_R, the interface wanders freely — roughness W ∝ L^ζ with ζ=1/2 (Edwards-Wilkinson universality). For the Ising model in 2D, T_R = T_c (the Onsager critical temperature). In 3D crystal surfaces (SOS model), T_R < T_c. The roughening transition is relevant to crystal growth, step dynamics, and polymer adsorption. The interface width W² = ⟨h²⟩ − ⟨h⟩² is shown in real time.