Kibble-Zurek mechanism and infinitely slow annealing through critical points

We revisit the Kibble-Zurek mechanism by analyzing the dynamics of phase ordering systems during an infinitely slow annealing across a second-order phase transition. We elucidate the time and cooling rate dependence of the typical growing length, and we use it to predict the number of topological defects left over in the symmetry broken phase as a function of time, both close and far from the critical region. Our results extend the Kibble-Zurek mechanism and reveal its limitations.

Figure 1

(Color online) (a) The control parameter, $\Delta g\left(t\right)=\left[g\left(t\right)-g_c\right]/g_c$, for different cooling rates. The crossover between adiabatic and out of equilibrium dynamics are signaled as $-{\widehat{t}}_i$ for ${\tau }_{Q_i}<{\tau }_{Q_{i+1}}$. (b) Sketch of the control parameter dependence of the equilibrium correlation length (thick red line) and the dynamic growing length $R$ for four linear cooling rate procedures with ${\tau }_{Q_i}<{\tau }_{Q_{i+1}}$. Values of the control parameter at which the dependence changes from adiabatic to critical are shown as ${\widehat{g}}_i$ (for simplicity we plot them as singular points in the evolution of $R$. In reality they just correspond to crossovers). For comparison the assumption of constancy during the impulse and subcritical regimes are shown with thin horizontal lines.

Figure 2

(Color online) Top panel: Test of dynamic scaling hypothesis (2) and the limits [Eq. (3)] in the $2d\text{IM}$ on a triangular and a square lattice, annealed at different cooling rates given in the key. A zoom on the critical region $\left|x\right|\lesssim 1$ is shown in the inset. The exponents are $\nu =1$, $z_{eq}=2.17$, and $z_d=2$. Bottom panel: number of defects at $t=t_Q$ for different cooling rates. Points represent the numerical data while the line corresponds to the prediction $N\left({\tau }_Q\right)={\tau }_Q^{-d/z_d}$