Kibble-Zurek Mechanism for Dynamical Ordering in a Driven Vortex System

The Kibble-Zurek mechanism describes the formation of topological defects in systems crossing a continuous symmetry-breaking phase transition at a finite quench rate. While this mechanism has been extensively studied for equilibrium transitions, its applicability to nonequilibrium transitions has not yet been fully examined. Recent simulation has shown the applicability of the Kibble-Zurek mechanism to dynamical ordering transitions in particlelike assemblies, including superconducting vortices, driven over random disorder. Here, we experimentally study the configurational order of vortices in the course of dynamical ordering with various quench rates. We verify a power-law scaling of the defect density with the quench rate and an impulse-adiabatic crossover on the ordered side of the transition, which are key predictions of the Kibble-Zurek mechanism. Our results suggest the applicability of the Kibble-Zurek mechanism to other nonequilibrium phase transitions.

Figure 1

(a) $I\text{−}V$ characteristics (driving force-velocity curves) (solid circles) and the differential resistance $dV/dI$ (open squares). (b) The initial value of the readout voltage $V_{\mathrm{read}}^0$, reflecting the degree of order of the vortex configuration after the quench from $I=0$ to $I_f$ with various quench rates $\dot{I}$. The values of $\dot{I}$ are listed on the right-hand side. The dotted line represents the approximate upper bound of $V_{\mathrm{read}}^0(I_f,\dot{I})$ and hence the most ordered configuration in the adiabatic limit, corresponding to $P_6=1$. Other lines are a guide to the eye.

Figure 2

(a) Schematics of the experimental protocol. (b) The readout voltage $V(t)$ induced by vortex motion in response to the square current for the vortex configuration after the quench at $I_f=3\mathrm{mA}$ with various $t_q$. The values of $t_q$ are shown in the figure.

Figure 3

(a) $P_6$ versus $I_f$ deduced from the data in Fig. 1. The symbols with different colors correspond to those in Fig. 1. The lines are a guide to the eye. (b) The defect density $P_D(=1-P_6)$ versus ${\dot{I}}^{-1}$ for $I_f=3$ (solid circles) and 2.2 mA (open circles) plotted in a double logarithmic scale. The solid and dashed lines are the power-law fits $P_D\propto {\dot{I}}^{\beta }$ with $\beta =0.5\pm 0.06$ and $\beta =0.42_{-0.14}^{+0.06}$, respectively. The dotted lines are a guide to the eye.

Figure 4

(a) The difference of $P_6(I_f)$ for various $\dot{I}$ ($\ge 3\times {10}^1\mathrm{mA}/\mathrm{s}$) from the adiabatic limit $P_6^{\mathrm{ad}}(I_f)$. Here, we use $P_6$ for the smallest quench rate $\dot{I}=3\times {10}^{-1}\mathrm{mA}/\mathrm{s}$ as the adiabatic limit $P_6^{\mathrm{ad}}$. The lines are a guide to the eye. (b) The log-log plot of $(I_f^{\mathrm{peak}}-I_1)/I_1$ versus ${\dot{I}}^{-1}$. $I_f^{\mathrm{peak}}$ is a current at which $P_6^{\mathrm{ad}}-P_6$ in (a) takes a peak or maximum value, and $I_1(=0.56\mathrm{mA})$ is a threshold current separating the plastic flow and the smectic flow regimes. The straight line represents the power-law fit $(I_f^{\mathrm{peak}}-I_1)/I_1\propto {\dot{I}}^{1/(1+\nu z)}$ with $\nu z=1.7\pm 0.3$.