Sequential Kibble-Zurek dynamics in the anisotropic Ising model of the Si(001) surface

As a simplified description of the non-equilibrium dynamics of buckled dimers on the Si(001) surface, we consider the anisotropic 2D Ising model and study the freezing of spatial correlations during a cooling quench across the critical point. The dependence of the frozen correlation lengths ξ‖ and ξ⊥ on the cooling rate obtained numerically matches the Kibble-Zurek scaling quite well. However, we also find that the ratio ξ‖/ξ⊥ of their frozen values deviates significantly from the ratio in equilibrium. Supported by analytical arguments, we explain this difference by the fact that the deviation from equilibrium in the weakly coupled direction occurs earlier than in the strongly coupled direction.