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arXiv:2609.35485

Out-of-equilibrium finite-time percolation transitions and spinodal-like behaviors after quenches across magnetic first-order transitions of Ising systems

Andrea Pelissetto C, Davide Rossini, Ettore Vicari

Published Sep 28, 2026• 0 citations
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Abstract

We analyze the out-of-equilibrium relaxational dynamics of ferromagnetic Ising-like systems driven across their low-temperature magnetic first-order transition (FOT) line, by sudden and slow variations of an external homogenous magnetic field $h$. As a paradigmatic example, we consider a two-dimensional Ising system driven across its low-temperature FOT line by suddently quenching $h$ from $h_i<0$ to $h>0$, inducing a transition from the negatively to the positively magnetized phase. We show the emergence of a dynamic percolation transition at a finite critical time $t_c(h)$ along the post-quench evolution for finite (sufficiently small) values of $h$, which is marked by the percolation of the largest positive-spin cluster and the antipercolation of the largest negative-spin cluster. 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 transition marks the passage from the metastable negatively-magnetized phase to the stable positively-magnetized one. Therefore, in the small-$h$ limit, the percolation critical behavior is related to the spinodal-like behavior of the magnetization, implying that the percolation time $t_c(h)$ exhibits a spinodal-like exponential dependence on $h$. The existence of percolation transitions is likely a generic phenomenon at magnetic Ising-like FOTs. For example, we observe an analogous behavior in dynamic (Kibble-Zurek-like) protocols entailing slow variations of the magnetic field driving the crossing of the FOT line.

DOI
10.48550/arxiv.2609.35485

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We analyze the out-of-equilibrium relaxational dynamics of ferromagnetic Ising-like systems driven across their low-temperature magnetic first-order transition (FOT) line, by sudden and slow variations of an external homogenous magnetic field hhh. As a paradigmatic example, we consider a two-dimensional Ising system driven across its low-temperature FOT line by suddently quenching hhh from hi<0h_i<0hi​<0 to h>0h>0h>0, inducing a transition from the negatively to the positively magnetized phase. We show the emergence of a dynamic percolation transition at a finite critical time tc(h)t_c(h)tc​(h) along the post-quench evolution for finite (sufficiently small) values of hhh, which is marked by the percolation of the largest positive-spin cluster and the antipercolation of the largest negative-spin cluster. 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 hhh. We also show that the percolation transition marks the passage from the metastable negatively-magnetized phase to the stable positively-magnetized one. Therefore, in the small-hhh limit, the percolation critical behavior is related to the spinodal-like behavior of the magnetization, implying that the percolation time tc(h)t_c(h)tc​(h) exhibits a spinodal-like exponential dependence on hhh. The existence of percolation transitions is likely a generic phenomenon at magnetic Ising-like FOTs. For example, we observe an analogous behavior in dynamic (Kibble-Zurek-like) protocols entailing slow variations of the magnetic field driving the crossing of the FOT line.
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 hhh from hi<0h_i<0hi​<0 to h>0h>0h>0. In the thermodynamic limit and for finite values of hhh, the post-quench evolution under a purely relaxational dynamics is characterized by a dynamic transition at a finite critical time tc(h)t_c(h)tc​(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 hhh. We also show that the percolation critical behavior is related to the spinodal-like behavior of the magnetization in the small-hhh limit, which implies that the percolation time tc(h)t_c(h)tc​(h) exhibits a spinodal-like exponential dependence on hhh.
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.
We study the out-of-equilibrium dynamics of one-dimensional quantum Ising models in a transverse field ggg, driven by a time-dependent longitudinal field hhh across their {\em magnetic} first-order quantum transition at h=0h=0h=0, for sufficiently small values of ∣g∣|g|∣g∣. We consider nearest-neighbor Ising chains of size LLL with periodic boundary conditions. We focus on the out-of-equilibrium behavior arising from Kibble-Zurek protocols, in which hhh is varied linearly in time with time scale tst_sts​, i.e., h(t)=t/tsh(t)=t/t_sh(t)=t/ts​. The system starts from the ground state at hi≡h(ti)<0h_i\equiv h(t_i)<0hi​≡h(ti​)<0, where the longitudinal magnetization MMM is negative. Then it evolves unitarily up to positive values of h(t)h(t)h(t), where M(t)M(t)M(t) becomes eventually positive. We identify several scaling regimes characterized by a nontrivial interplay between the size LLL and the time scale tst_sts​, which can be observed when the system is close to one of the many avoided level crossings that occur for h≥0h\ge 0h≥0. In the L→∞L\to\inftyL→∞ limit, all these crossings approach h=0+h=0^+h=0+, making the study of the thermodynamic limit, defined as the limit L→∞L\to\inftyL→∞ keeping ttt and tst_sts​ constant, problematic. We study such limit numerically, by first determining the large-LLL quantum evolution at fixed tst_sts​, and then analyzing its behavior with increasing tst_sts​. Our analysis shows that the system switches from the initial state with M<0M<0M<0 to a positively magnetized state at h=h∗(ts)>0h = h_*(t_s)>0h=h∗​(ts​)>0, where h∗(ts)h_*(t_s)h∗​(ts​) decreases with increasing tst_sts​, apparently as h∗∼1/ln⁡tsh_*\sim 1/\ln t_sh∗​∼1/lnts​. This suggests the existence of a scaling behavior in terms of the rescaled time Ω=tln⁡ts/tsΩ= t \ln t_s/t_sΩ=tlnts​/ts​. The numerical results also show that the system converges to a nontrivial stationary state in the large-ttt limit, characterized by an energy significantly larger than that of the corresponding homogeneously magnetized ground state.
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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 T0T0T0 of the magnetic field, which decrease as h∗∼1/(ln⁡ts)κh_* \sim 1/(\ln t_s)^κh∗​∼1/(lnts​)κ, with κ=2κ= 2κ=2 and κ=1κ=1κ=1 in two and three dimensions, respectively, for ts→∞t_s\to\inftyts​→∞. We identify σ≡t(ln⁡t)κ/tsσ\equiv t (\ln t)^κ/t_sσ≡t(lnt)κ/ts​ as the relevant scaling variable associated with the KZ dynamics in the TL.
We study the out-of-equilibrium spinodal-like dynamics of three-dimensional qqq-state Potts systems driven across their thermal first-order transition in the thermodynamic limit, by a relaxational (heat-bath) dynamics. During the evolution, the inverse temperature βββ increases linearly with time, as δβ(t)≡β(t)−βfo∼t/tsδβ(t)\equiv β(t)- β_{\rm fo} \sim t/t_sδβ(t)≡β(t)−βfo​∼t/ts​, where βfoβ_{\rm fo}βfo​ is the inverse temperature at the transition point, ttt is the time and tst_sts​ is a time scale. The dynamics starts at ti<0t_i< 0ti​<0 from an ensemble of disordered configurations equilibrated at inverse temperature β(ti)<βfoβ(t_i)<β_{\rm fo}β(ti​)<βfo​ and ends at positive values of ttt, when the system is ordered (this is analogous to a standard Kibble-Zurek protocol). The time-dependent energy density shows an out-of-equilibrium scaling behavior in the large-tst_sts​ limit, in terms of the scaling variable t(ln⁡t)κ/tst(\ln t)^κ/t_st(lnt)κ/ts​. The corresponding exponent turns out to be consistent with κ=3/2κ=3/2κ=3/2 (with a good accuracy), which is the value obtained by assuming that the initial nucleation of ordered regions provides the relevant mechanism for the passage from one phase to the other. The scaling behavior implies a spinodal-like phenomenon close to the transition point: the passage from the disordered to the ordered phase, composed of large ordered regions of different color, occurs at δβ(t)=δβ∗>0δβ(t)=δβ_*>0δβ(t)=δβ∗​>0, where δβ∗δβ_*δβ∗​ decreases as 1/(ln⁡ts)3/21/(\ln t_s)^{3/2}1/(lnts​)3/2 in the large-tst_sts​ limit.