Out-of-equilibrium spinodal-like scaling behaviors across the magnetic first-order transitions of two-dimensional and three-dimensional 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 magnetic first-order transitions at low temperature (Formula presented), where (Formula presented) is the temperature of their continuous transition. We consider Kibble-Zurek (KZ) protocols in which a spatially homogenous magnetic field (Formula presented) varies as (Formula presented) with a timescale (Formula presented). The KZ dynamics starts from negatively magnetized configurations equilibrated at (Formula presented) and stops at a positive value of (Formula presented) 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 (Formula presented) and the timescale (Formula presented) diverge simultaneously, keeping an appropriate combination fixed. Then, we analyze the KZ dynamics in the thermodynamic limit (TL), obtained by taking first the (Formula presented) limit at fixed (Formula presented) and (Formula presented), and then considering the scaling behavior in the large-(Formula presented) 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 (Formula presented) of the magnetic field, which decrease as (Formula presented), with (Formula presented) and (Formula presented) in two and three dimensions, respectively, for (Formula presented). We identify (Formula presented) as the relevant scaling variable associated with the KZ dynamics in the TL.