Dynamics of the order parameter in symmetry breaking phase transitions

Abstract

The formation of topological defects in second-order phase transitions can be investigated by solving partial differential equations for the evolution of the order parameter in space and time, such as the Langevin equation. We demonstrate that the ordinary differential equations governing either the temporal or spatial dependence in the Langevin equation provide surprisingly substantial insights into the dynamics of the phase transition. The temporal evolution of the order parameter predicts the essence of the adiabatic-impulse scenario, including the scaling of the freeze-out time, which is crucial to the Kibble-Zurek mechanism (KZM). In particular, Bernoulli differential equations that arise in the overdamped case can be solved analytically. The spatial part of the evolution, in turn, leads to the characteristic size of domains that choose the same broken symmetry. Apart from the fundamental insights into the KZM, this finding enables the exploration of Kibble-Zurek scaling using ordinary differential equations over a large range of quench timescales, which would otherwise be difficult to achieve with numerical simulations of the full partial differential equations.

Figures (5)

• Figure 1: (a) The order parameter $\varphi (t)$, Eq. (\ref{['solu']}), for $\eta=1$, $\tau_Q=128$, and $\varphi (0)=10^{-2}, 10^{-3}, 10^{-4}, 10^{-5}$, from thick to thin line, as well as the numerical solutions (see Fig. \ref{['fig2']}). The grey line is equilibrium $|\varphi_{\rm min}|=\sqrt{\epsilon}$. The thick and thin purple vertical lines indicate $\pm \hat{t}$ for $\varphi (0)=10^{-2}$ and $10^{-5}$ respectively. The numerical results $\sqrt{\langle \Phi (x,t)^2\rangle}$ (solid red), $\hbox{max}_x |\Phi (x,t)|$ (dashed red) are obtained from Eq. (\ref{['langevin']}) with $\eta=1$ and $\theta=10^{-4}$. (b) Plots of $\log \varphi(t)$ for $\varphi (0)=10^{-2} ...10^{-8}$. As $\varphi (t) \approx {\varphi (0)} e^{t^2/4\eta\tau_Q}$ when $t \in [-\hat{t},+\hat{t}]$, perturbations present at $-\hat{t}$ reappear at $+\hat{t}$, so they are in effect "frozen". Noise (see Eq. (1)) added when $t \in [-\hat{t},+\hat{t}]$ is amplified, but the freeze-out time $+\hat{t}$ is insensitive (depends logarithmically) on $\varphi(0)$.
• Figure 2: 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 $\sqrt{\langle \Phi (x,t)^2\rangle}$ (solid 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: The rescaled solution $\tilde{\varphi} (\tilde{t})$ in the overdamped case with $\eta=1$ where the first term of Eq. (\ref{['ode1']}) is discarded (a) and in the underdamped case where the second term of Eq. (\ref{['ode1']}) is discarded (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 minimum of the potential $V$.
• Figure 4: The freeze-out time $\hat{t}$ as a function of the quench timescale $\tau_Q$ for damping constants (a) $\eta = 0.01$, (b) $\eta = 0.2$, and (c) $\eta=1$, with $\varphi (0)=10^{-4}$. The color plot represents the slope of the log plot based on nearest neighbor points. For $\eta = 0.2$, a transition from the underdamped regime to the overdamped regime is observed as $\tau_Q$ increases. The dashed gray line represents the theoretical prediction $\tau_Q = 1/\eta^3$ where the transition occurs. (d) The number of defects $\mathcal{N}$ as the function of $\tau_Q$ for $\eta=0.01$ (dashed blue line), $\eta=0.2$ (solid blue line for $\tau_Q\in[4,128]$ and solid red line for $\tau_Q\in[128,16384]$), and $\eta=1$ (thick red line) with $\theta=10^{-8}$.
• Figure 5: (a) $s(t,t_F)$ for $\tau_Q=128$ (thick, red) and $\tau_Q=512$ (blue). $t_F=300$ and $\theta=10^{-8}$. Vertical dashed lines indicate $\pm \hat{t}$. (b,c) $\langle \lambda (t) \rangle$ as a function of $\tau_Q$ for $\eta=1$ (b) and for $\eta=0.01$ (c). From thick red to thin black line, $t=-2\hat{t}, -\hat{t}, 0, \hat{t}/2, \hat{t}$ respectively. The dashed black lines represent $L/\mathcal{N}$ (i.e., the average domain size) from Fig. \ref{['fig4']} (d). $\theta=10^{-8}$.