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arXiv:2608.05936

Dynamic scaling behavior in the presence of a periodic magnetic driving across Ising continuous transitions

Andrea Pelissetto C, Ettore Vicari

Published Aug 6, 2026• 0 citations
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Abstract

We study the critical dynamics arising from a time-dependent periodic homogenous source coupled to the order-parameter field, which drives a classical ferromagnetic system across a continuous transition. For this purpose, we consider the paradigmatic two-dimensional (2D) Ising model in the presence of a periodic magnetic field $h(t)=-A\, \cos (2πt/P)$, evolving under a purely relaxational dynamics at the critical temperature. We show that the periodic driving gives rise to a peculiar dynamic scaling behavior in the thermodynamic limit, arising from a nontrivial interplay among the time $t$, the amplitude $A$ and period $P$ of $h(t)$. The relevant scaling variables are $τ=t/P$ and $σ=A P^κ$, with $κ= y_h/z$, where $y_h=(d+2-η)/2$ is the critical dimension of the magnetic field, and $z$ is dynamic exponent for the critical relaxational dynamics ($κ\approx 0.865$ for the 2D Ising model). The dynamic scaling behaviors of the magnetization and bond-energy density show an oscillatory behavior around a smooth curve which approaches a large-$τ$ stationary behavior. We also briefly discuss the dynamic behavior of an Ising system driven across the critical point by a periodic time-varying temperature at zero magnetic field.

DOI
10.48550/arxiv.2608.05936

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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.
Abstract When a ferromagnetic system is driven by an oscillating magnetic field, the system shows various intriguing magnetic properties. The two most common features are dynamic phase transition and dynamic hysteresis resulting from the competition between the period of the external field and the metastable lifetime of the system. In this article, we mainly review the similarities and dissimilarities between ferromagnetic systems in thermal equilibrium and those exposed to a driving time-dependent magnetic field. Extensive studies performed over two decades have shown that dynamic and thermodynamic phase transitions correspond to the same universality class and have similar phase diagrams, among other similarities. Remarkably, recent experimental studies have revealed that both fluctuations in dynamic order parameter and susceptibility can demonstrate anomalously sharp sidebands, which are known as metamagnetic anomalies, in the dynamically paramagnetic phase close to the critical point. This behavior is one of the most interesting behaviors observed in the magnetic systems driven by a time-dependent field which does not have an analog in the corresponding equilibrium counterparts. We discuss the general picture regarding the effects of the applied field components on the metamagnetic anomalies. As a main conclusion, we remark that a comprehensive literature overview reveals that the similarities between conventional thermodynamic phase transitions and corresponding dynamic counterpart are only limited to the vicinity of critical period regime with zero bias field, and they should be treated very carefully in the presence of surfaces. Finally, some of the main questions waiting to be answered and potential future research directions are underlined.
The dynamic phase transitions of periodically driven ferromagnets present a complex interplay between internal relaxation dynamics and external driving forces. In this study, a highly optimized Metropolis-Hastings Monte Carlo simulation is developed to model a two-dimensional Ising ferromagnet subjected to an oscillating magnetic field. Following exact thermodynamic validation of the static critical exponent of 1.75, the system is pushed out of equilibrium to investigate two competing mechanisms of phase disruption: structural spatial disorder and temporal frequency variation. Computational results demonstrate that quenched non-magnetic impurities act as localized pinning centers for expanding domain walls, fundamentally "softening" the material and drastically shrinking the dynamically ordered parameter space. Conversely, scaling the driving frequency outpaces localized spin relaxation, effectively bypassing the chaotic limit cycle and expanding the dynamically ordered phase. This dichotomy offers a comprehensive mathematical profile of non-equilibrium phase boundaries in disordered magnetic lattices.
We study the equilibrium and dynamic phase transition properties of two-dimensional Ising model on a decorated triangular lattice under the influence of a time-dependent magnetic field composed of a periodic square wave part plus a time independent bias term. Using Monte Carlo simulations with standard Metropolis algorithm, we determine the equilibrium critical behavior in zero field. At a fixed temperature corresponding to the multidroplet regime, we locate the relaxation time and the dynamic critical half-period at which a dynamic phase transition takes place between ferromagnetic and paramagnetic states. Benefiting from finite-size scaling theory, we estimate the dynamic critical exponent ratios for the dynamic order parameter and its scaled variance, respectively. The response function of the average energy is found to follow a logarithmic scaling as a function of lattice size. At the critical half-period and in the vicinity of small bias field regime, average of the dynamic order parameter obeys a scaling relation with a dynamic scaling exponent which is very close to the equilibrium critical isotherm value. Finally, in the slow critical dynamics regime, investigation of metamagnetic fluctuations in the presence of bias field revels a symmetric double-peak behavior for the scaled variance contours of dynamic order parameter and average energy. Our results strongly resemble those previously reported for kinetic Ising models.
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.