Dynamic scaling of topological ordering in classical systems*

We analyze scaling behaviors of simulated annealing carried out on various classical systems with topological order, obtained as appropriate limits of the toric code in 2D and 3D. We first consider the 3D Ising lattice gauge model, which exhibits a continuous topological phase transition at finite temperature. We show that a generalized Kibble-Zurek scaling ansatz applies to this transition, in spite of the absence of a local order parameter. We find perimeter-law scaling of the magnitude of a non-local order parameter (defined using Wilson loops) and a dynamic exponent z = 2.70 ± 0.03. We then study systems where (topological) order forms only at zero temperature—the Ising chain, the 2D Ising gauge model, and a 3D star model (another variant of the 3D Ising gauge model). We show that the Kibble-Zurek theory does not apply in any of these systems. Instead, the dynamics can be understood in terms of diffusion and annihilation of topological defects, which we use to formulate a scaling theory in good agreement with our simulation results. We also discuss the effect of open boundaries where defect annihilation competes with a faster process of evaporation at the surface.