Deconstructing symmetry breaking dynamics

Abstract

The Kibble-Zurek mechanism (KZM) successfully predicts the density of topological defects deposited by the phase transitions, but it is not clear why. Its key conjecture is that, near the critical point of the second-order phase transition, critical slowing down will result in a period when the system is too sluggish to follow the potential that is changing faster than its reaction time. The correlation length at the freeze-out instant $\hat t$ when the order parameter catches up with the post-transition broken symmetry configuration is then decisive, determining when the mosaic of broken symmetry domains locks in topological defects. To understand why the KZM works so well we analyze Landau-Ginzburg model and show why temporal evolution of the order parameter plays such a key role. The analytical solutions we obtain suggest novel, hitherto unexplored, experimentally accessible observables that can shed light on symmetry breaking dynamics while testing the conjecture on which the KZM is based.

Figures (8)

• Figure 1: Symmetry breaking dynamics in a second order phase transition. Landau-Ginzburg model attributes symmetry breaking to the change of the thermodynamic potential from the single-minimum parabola (top) to the degenerate minima (bottom). When the order parameter is a real field (case represented by the dark line above) there are just two broken symmetry minima. When the field is complex (i.e., order parameter in a superfluid), the potential assumes a "sombrero’’ shape, with continuum of the broken symmetry states. We investigate how the temporal evolution of the order parameter (which starts for $t<0$ in the single symmetric minimum, but must choose one of the broken symmetry possibilities after the transition, see Fig. \ref{['fig3']}) leads to the mosaic of domains that lock in topological defects. The Kibble-Zurek mechanism posits that the size of these domains is set by the correlation length at the instant $+\hat{t}$ when the order parameter catches up with the new broken symmetry minimum. We show that this temporal evolution can be analyzed using ordinary differential equations with solutions that mark the freeze-out instant $+\hat{t}$.
• Figure 2: Time evolution of the order parameter: comparison between the analytical solution of the ordinary differential equation (\ref{['ode1']}) and the numerical solution of the Langevin equation for a real scalar field in (1+1) dimensions [\ref{['langevin']}]. The thick lines represent the analytical solutions $\varphi (t)$ (Eq. (\ref{['solu']})) with $\eta=1$ and $\varphi (0)=10^{-4}$ for various quench timescales $\tau_Q$, while the numerical results $\sqrt{\langle \Phi (x,t)^2\rangle}$ (thin lines), $\hbox{max}_x |\Phi (x,t)|$ (dashed lines) are obtained by solving Eq. (\ref{['langevin']}) with $\eta=1$ and $\theta=10^{-8}$. From left to right, $\tau_Q=128,256,512,1024$ respectively.
• Figure 3: Order parameter dynamics for varying noise temperature $\theta$ and initial conditions $\varphi (0)$. (a) The order parameter $\varphi(t)$ from Eq. (\ref{['solu']}) for $\eta=1$, $\tau_Q=128$, and $\varphi(0) = 10^{-2}$, $10^{-3}$, $10^{-4}$, $10^{-5}$ (thick to thin lines), compared with numerical results: $\sqrt{\langle \Phi^2 \rangle}$ (solid red) and $\max_x |\Phi|$ (dashed red) from Eq. (\ref{['langevin']}) for a real scalar field in (1+1) dimensions with $\eta=1$, $\theta=10^{-4}$ (see Fig. \ref{['fig2']}). The gray line shows equilibrium $|\varphi_{\rm min}| = \sqrt{\epsilon}$. Vertical purple lines mark $\pm \hat{t}$ for $\varphi(0)=10^{-2}$ (thick) and $10^{-5}$ (thin). (b) Noise-induced $\sqrt{\langle \Phi^2 \rangle}$ with $\theta= 10^{-2}$ to $10^{-5}$ (thick to thin) slowly accumulates before $t = 0$, setting $\varphi(0)\approx \sqrt{\langle \Phi (x,t=0)^2\rangle} \approx\frac{1}{2} \sqrt{2\theta /\hat{\epsilon}}$ (dashed lines) for rapid post-transition growth. (c) Plots of $\log \varphi(t)$ for $\varphi (0)=10^{-1} ...10^{-8}$. As $\varphi (t) \approx {\varphi (0)} e^{t^2/4\eta\tau_Q}$ when $t \in [-\hat{t},+\hat{t} ]$, order parameter buffeted by noise is amplified, but the freeze-out time $+\hat{t}$ depends only logarithmically on $\varphi(0)$, making it insensitive to its precise value, and, hence, insensitive to $\theta$, the noise temperature.
• Figure 4: Order parameter evolution for the Gross–Pitaevskii equation. The black line represents the analytical solution for the condensate number density $|\varphi (t)|^2$ (Eq. (\ref{['solu2']})) with $\varphi (0)=3\times 10^{-3}$, while the corresponding numerical result $\langle |\Phi (x,t)|^2\rangle$ (light red dots) is obtained by solving the stochastic Gross–Pitaevskii equation Eq. (\ref{['gpe']}) in (1+1) dimensions for a complex wave function-like order parameter with $\theta=10^{-3}$. For both plots, $\gamma=10^{-2}$, $g=0.05$, and quench timescale $\tau_Q=0.05$.
• Figure 5: Order parameter dynamics given by Eq. (\ref{['ode1']}) in the overdamped and underdamped cases corresponding to a real scalar field described by the Langevin equation. The rescaled solution $\tilde{\varphi} (\tilde{t})$ in the overdamped case with $\eta=1$ where the first term $\ddot{\varphi}$ of Eq. (\ref{['ode1']}) is discarded (a) and in the underdamped case where the second term $\eta \dot{\varphi}$ of Eq. (\ref{['ode1']}) is discarded (b). $\tilde{\varphi} = \sqrt[4]{\tau_Q/\eta} \varphi$ and $\tilde{t} = t / \sqrt{\eta \tau_Q}$ in (a). $\tilde{\varphi}=\tau_Q^{1/3}\varphi$, $\tilde{t}= t/\tau_Q^{1/3}$ in (b). $\varphi (0)=10^{-4}$ and various quench timescales $\tau_Q$. The dashed line represents $\tilde{\varphi}=\sqrt{\epsilon (\tilde{t})}$, the location of the broken symmetry minimum of the potential $V$.