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Out-of-equilibrium spinodal-like scaling behaviors across the magnetic first-order transitions of 2D and 3D Ising systems

We study the out-of-equilibrium scaling behavior of two-dimensional and three-dimensional Ising systems, when they are slowly driven across their {\em magnetic} first-order transitions at low temperature $T<T_c$, where $T_c$ is the temperature of their continuous transition. We consider Kibble-Zurek (KZ) protocols in which a spatially homogenous magnetic field $h$ varies as $h(t)=t/t_s$ with a time scale $t_s$. The KZ dynamics starts from negatively-magnetized configurations equilibrated at $h_i<0$ and stops at a positive value of $h$ where the configurations acquire a positive average magnetization. We consider the Metropolis and the heat-bath dynamics, which are two specific examples of a purely relaxational dynamics. We focus on two different dynamic regimes. We consider the out-equilibrium finite-size scaling (OFSS) limit in which the system size $L$ and the time scale $t_s$ diverge simultaneously, keeping an appropriate combination fixed. Then, we analyze the KZ dynamics in the thermodynamic limit (TL), obtained by taking first the $L\to\infty$ limit at fixed $t$ and $t_s$, and then considering the scaling behavior in the large-$t_s$ limit. Our numerical results provide evidence of OFSS, as predicted by general scaling arguments. The results in the TL show the emergence of spinodal-like behaviors: The passage from the negatively-magnetized phase to the positively-magnetized one occurs at positive values $h_*>0$ of the magnetic field, which decrease as $h_* \sim 1/(\ln t_s)^κ$, with $κ= 2$ and $κ=1$ in two and three dimensions, respectively, for $t_s\to\infty$. We identify $σ\equiv t (\ln t)^κ/t_s$ as the relevant scaling variable associated with the KZ dynamics in the TL.