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arXiv:2603.13127

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

Andrea Pelissetto C, Davide Rossini, Ettore Vicari

Published Mar 13, 2026• 0 citations
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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 $h$.

DOI
10.48550/arxiv.2603.13127

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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 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 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 explore the connection between the low-temperature boundary of the ferromagnetic phase in the two-dimensional ±J\pm J±J random-bond Ising model, where antiferromagnetic bonds occur with probability ppp and a geometric transition dubbed ``zero-weight percolation''. We argue that the onset of this percolation characterized by the emergence of a percolating path containing an equal number of +J+J+J and −J-J−J bonds is incompatible with ferromagnetic ordering. Due to its purely geometrical nature, this percolation criterion is a property of a disorder realization and is independent of the temperature, which in turn suggests that the ferromagnetic phase boundary is vertical below the Nishimori point in the (p,T)(p,T)(p,T) plane. Using a dynamic-programming algorithm combined with finite-size scaling, we identify the critical disorder at which zero-weight paths first percolate as pc=0.1000(2)p_c = 0.1000(2)pc​=0.1000(2), and we extract the associated critical exponents ν=1.26(1)ν= 1.26(1)ν=1.26(1), β/ν=0.85(1)β/ν= 0.85(1)β/ν=0.85(1), γ/ν=0.264(5)γ/ν= 0.264(5)γ/ν=0.264(5), and fractal dimension df≈1.11d_f \approx 1.11df​≈1.11. The value of pcp_cpc​ is below the previously reported values of the critical disorder strength corresponding to the loss of the ferromagnetic order, both at zero temperature and the Nishimori point. Nevertheless, we argue that the percolation transition studied in this paper is behind the loss of ferromagnetism and thus provides a new, purely geometrical perspective on the stability of ferromagnetic order in disordered spin 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 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.
The ground-state structure of the two-dimensional random field Ising magnet is studied using exact numerical calculations. First we show that the ferromagnetism, which exists for small system sizes, vanishes with a large excitation at a random field strength-dependent length scale. This breakup length scale L(b) scales exponentially with the squared random field, exp(A/delta(2)). By adding an external field H, we then study the susceptibility in the ground state. If L>L(b), domains melt continuously and the magnetization has a smooth behavior, independent of system size, and the susceptibility decays as L-2. We define a random field strength-dependent critical external field value +/-H(c)(delta) for the up and down spins to form a percolation type of spanning cluster. The percolation transition is in the standard short-range correlated percolation universality class. The mass of the spanning cluster increases with decreasing Delta and the critical external field approaches zero for vanishing random field strength, implying the critical field scaling (for Gaussian disorder) H(c) approximately (delta-delta(c))(delta), where delta(c)=1.65+/-0.05 and delta=2.05+/-0.10. Below Delta(c) the systems should percolate even when H=0. This implies that even for H=0 above L(b) the domains can be fractal at low random fields, such that the largest domain spans the system at low random field strength values and its mass has the fractal dimension of standard percolation D(f)=91/48. The structure of the spanning clusters is studied by defining red clusters, in analogy to the "red sites" of ordinary site percolation. The sizes of red clusters define an extra length scale, independent of L.
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.