Out-of-equilibrium percolation transitions at finite critical times after quenches across magnetic first-order transitions

Abstract

We show that an out-of-equilibrium percolation transition occurs after quenching ferromagnetic Ising-like systems across their magnetic first-order transitions. As a paradigmatic example, we consider a two-dimensional Ising system driven across its low-temperature first-order transition line by a quench of the magnetic field $h$ from $h_i<0$ to $h>0$. In the thermodynamic limit and for finite values of $h$, the post-quench evolution under a purely relaxational dynamics is characterized by a dynamic transition at a finite critical time $t_c(h)$ from the metastable negatively magnetized phase to the positive one, marked by the percolation of the largest clusters of positive and negative spins. This out-of-equilibrium percolation transition displays a finite-size scaling behavior as in the standard random-percolation case. However, while the fractal dimension of the percolating clusters is consistent with the random-percolation value, the exponent controlling the approach to criticality differs and depends on $h$. We also show that the percolation critical behavior is related to the spinodal-like behavior of the magnetization in the small-$h$ limit, which implies that the percolation time $t_c(h)$ exhibits a spinodal-like exponential dependence on $h$.

Figures (9)

• Figure 1: The rescaled average sizes $\Sigma_+$ (bottom) and $\Sigma_-$ (top) of the largest clusters of positive and negative magnetization, defined in Eq. \ref{['drddef']}, as a function of the post-quench time $t$, for $h=0.07$ and several sizes $L$. Statistical errors are not visible, as they are smaller than the symbol size.
• Figure 2: Snapshots of the configurations for $h = 0.07$ and $L=1000$, at times $t = 1144$ (left), $t = t_c = 1174$ (center), and $t = 1204$ (right). Violet and green sites belong to the largest clusters of positive and negative spins, respectively; white areas correspond to smaller (positive and negative) clusters. The size of the largest positive cluster is $368001$ for $t = 1144$, $498264$ for $t=t_c=1174$, and $637204$ for $t=1204$. Its growth is due to the merging with smaller clusters, since the boundaries of the clusters are essentially unchanged.
• Figure 3: The time dependence of the ratio $R_s$ for $h = 0.07$. The vertical dashed line corresponds to the critical time $t_c = 1174$. The inset shows $R_s$ vs $X = (t-t_c) L^{w}$, with $w = 0.68$. The excellent collapse of the curves (in particular for $L > 600$) supports the Ansatz \ref{['Ansatz-FSS']}, confirming the equality $d_+=d_-$ of the fractal dimensions of positive and negative clusters.
• Figure 4: Plot of $a(h) L^{\delta_{\rm RP}} \Sigma_+$ vs $R_s$ for various $h$, where $a(h)$ is chosen to optimize the collapse of the data: Setting $a = 1$ for $h=0.045$, we use $a=1.137,1.096,1.037$ for $h=0.07,0.06,0.05$, respectively. We report results for $L = 2000$ (empty symbols) and $L=4000$ (filled symbols). The resulting scaling is good, with small deviations that can be explained by residual scaling corrections.
• Figure 5: The magnetization $M(t)$ plotted vs $\sigma = h(\ln t)^2$. The size $L$ is large enough that the data can be regarded in their thermodynamic limit. The inset shows the ratio $M(t)/M_0$ vs $\hat{\sigma}=(\sigma-\sigma_*) h^{-\theta}$ with $\theta=0.64$ and $\sigma_*= 3.03$, where $M_0\approx 0.940259$ is the spontaneous magnetization for $\beta = 1.2\beta_c$.