Universal Breakdown of Kibble-Zurek Scaling in Fast Quenches across a Phase Transition

The crossing of a continuous phase transition gives rise to the formation of topological defects described by the Kibble-Zurek mechanism (KZM) in the limit of slow quenches. The KZM predicts a universal power-law scaling of the defect density as a function of the quench time. We focus on the deviations from KZM experimentally observed in rapid quenches and establish their universality. While KZM scaling holds below a critical quench rate, for faster quenches the defect density and the freeze-out time become independent of the quench rate and exhibit a universal power-law scaling with the final value of the control parameter. These predictions are verified in several paradigmatic scenarios in both the classical and quantum domains.

Figure 1

Universal breakdown of KZM at fast quenches. The KZM relies on the adiabatic impulse approximation to determine the freeze-out time $\widehat{t}$, by equating the time elapsed after crossing the phase transition to relaxation time. For slow quenches, $\widehat{t}$ is given by the power law [Eq. (2)]. For fast quenches of finite depth ${\lambda }_f$, it is set by the equilibrium relaxation time with $\widehat{t}\sim \tau ({\lambda }_f)$, and becomes independent of the quench time. Thus, the density of defects saturates at a plateau, leading to the breakdown of the KZM scaling relations at fast quenches.

Figure 2

Universal breakdown of KZM at fast quenches in a one-dimensional ${\varphi }^4$ model in Log-Log plots. (a) The dependence of the density on the quench time is shown for three different values of ${\lambda }_f=-2,-1.5,-1$, from top to bottom. The inset shows a single realization of the scalar field ${\varphi }_l$, in the zigzag phase with excitations in the form kinks (at the interface of adjacent zigzag domains) after crossing the phase transition. In the KZM scaling regime, $n=(21.1\pm 0.4){\tau }_Q^{-0.245\pm 0.007}$, while the onset and value of the plateau depend on ${\lambda }_f$. The value of the density (b) and the freeze-out time (c) at the plateau, as well as the first critical quench rate ${\tau }_Q^{c1}$ (d), exhibit a universal power-law scaling with the quench depth. The plateau data are fitted by $n({\lambda }_f)=(2.53\pm 0.04){\epsilon }_f^{0.494\pm 0.005}$, $\widehat{t}=(628\pm 11){\epsilon }_f^{-0.95\pm 0.03}$, and ${\tau }_Q^{c1}=(5022\pm 57){\epsilon }_f^{-1.99\pm 0.02}$.

Figure 3

Universal deviations from the KZM in the 1D transversal field Ising model after a fast thermal quench. (a) The KZM power-law scaling of the kink density is interrupted by the onset of a plateau as the quench time is reduced. A fit to the data reads $P/L=(0.14\pm 0.03){\tau }_Q^{-0.54\pm 0.05}$ in the KZM regime. (b) In the fast quench case, the kink density is independent of the quench rate and scales universally with the quench depth ${\epsilon }_f$, with data fitted to $P/L=(0.177\pm 0.004){\epsilon }_f^{0.99\pm 0.05}$.

Figure 4

Universal deviations from the KZM in a 2D superfluid after a fast thermal quench. (a) The vortex number is independent of the quench rate and scales universally with the quench depth ${\epsilon }_f$. A fit to the data reads $n_{{\lambda }_f}=(1100\pm 5){\epsilon }_f^{0.99\pm 0.01}$. (b) Corresponding universal scaling of the freeze-out time as function of ${\epsilon }_f$, with data fitted to $\widehat{t}=(35.9\pm 0.2){\epsilon }_f^{-0.99\pm 0.03}$. The inset shows the distribution of vortices in a single realization.