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arXiv:2507.21286

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

Andrea Pelissetto C, Ettore Vicari

Published Jul 28, 2025• 0 citations
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Abstract

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 $T0$ 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.

DOI
10.48550/arxiv.2507.21286

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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 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 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.
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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.
We study the out-of-equilibrium Kibble-Zurek (KZ) dynamics in quantum Ising chains in a transverse field, driven by a time-dependent longitudinal field h(t)=t/tsh(t)=t/t_sh(t)=t/ts​ (tst_sts​ is the time scale of the protocol), across their first-order quantum transitions (FOQTs) at h=0h=0h=0. The KZ protocol starts at time ti<0t_i<0ti​<0 from the negatively magnetized ground state for hi=ti/ts<0h_i = t_i/t_s<0hi​=ti​/ts​<0. Then, the system evolves unitarily up to a time tf>0t_f > 0tf​>0, such that the magnetization of the state at time tft_ftf​ is positive. In finite-size systems, the KZ dynamics develops out-of-equilibrium finite-size scaling (OFSS) behaviors. Their scaling variables depend either exponentially or with a power law on the size, depending on the boundary conditions (BC). The OFSS functions can be computed in effective models restricted to appropriate low-energy (magnetized and/or kink) states. The KZ scaling behavior drastically changes in the thermodynamic limit (TL), defined as the infinite-size limit keeping ttt and tst_sts​ fixed, which appears substantially unrelated with the OFSS regime, because it involves higher-energy multi-kink states, which are irrelevant in the OFSS limit. The numerical analyses of the KZ dynamics in the TL show the emergence of a quantum spinodal-like scaling behavior at the FOQTs for all considered BC, which is independent of the BC. The longitudinal magnetization changes sign at h(t)=h∗>0h(t)=h*>0h(t)=h∗>0, where h∗h*h∗ decreases with increasing tst_sts​, as h∗∼1/ln⁡tsh*\sim 1/\ln t_sh∗∼1/lnts​. Moreover, in the large-tst_sts​ limit, the time-dependence of the magnetization is described by a universal function of Ω=t/τsΩ= t/τ_sΩ=t/τs​, with τs=ts/ln⁡tsτ_s = t_s/\ln t_sτs​=ts​/lnts​.
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