Dynamic scaling of topological ordering in classical systems

Abstract

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 two and three dimensions. We first consider the three-dimensional ${\mathbb{Z}}_{2}$ (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 nonlocal order parameter (defined using Wilson loops) and a dynamic exponent $z=2.70\ifmmode\pm\else\textpm\fi{}0.03$, the latter in good agreement with previous results for the equilibrium dynamics (autocorrelations). We then study systems where (topological) order forms only at zero temperature---the Ising chain, the two-dimensional ${\mathbb{Z}}_{2}$ gauge model, and a three-dimensional star model (another variant of the ${\mathbb{Z}}_{2}$ gauge model). In these systems the correlation length diverges exponentially, in a way that is nonsmooth as a finite-size system approaches the zero temperature state. 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.