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 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 timescale 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 timescale 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→∞ 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_{*}∼1/(lnt_{s})^{κ}, with κ=2 and κ=1 in two and three dimensions, respectively, for t_{s}→∞. We identify σ≡t(lnt)^{κ}/t_{s} as the relevant scaling variable associated with the KZ dynamics in the TL.