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arXiv:2607.05130

Critical behavior of the driven Curie-Weiss model

Ruohan Xu, Faezeh Khodabandehlou, Christian Maes

Published Jul 6, 2026• 0 citations
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Abstract

We complete the phase diagram of the macroscopic Curie-Weiss magnet in a time-periodic external field, as a function of temperature and driving parameters. There is a regime (large enough driving amplitude and frequency, at low temperatures) where stable paramagnetic and ferromagnetic phases coexist. In particular, we present a new detailed analysis of the (nonequilibrium) specific heat, diverging at the same critical inverse temperature $β_c$ as the magnetic susceptibility. The new Curie temperature decreases with the driving, and we find critical exponent $α=1$ for $β\downarrow β_c$, and $α\simeq 0.86$ for $β\uparrow β_c$, even for small driving. A Floquet analysis shows the nature of the criticality, which is dynamical, with implications that remain unseen and are mostly impossible when the system is in thermal equilibrium.

DOI
10.48550/arxiv.2607.05130

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We complete the phase diagram of the macroscopic Curie-Weiss magnet in a time-periodic external field, as a function of temperature and driving parameters. There is a regime (large enough driving amplitude and frequency, at low temperatures) where stable paramagnetic and ferromagnetic phases coexist. In particular, we present a new detailed analysis of the (nonequilibrium) specific heat, diverging at the same critical inverse temperature βcβ_cβc​ as the magnetic susceptibility. The new Curie temperature decreases with the driving, and we find critical exponent α=1α=1α=1 for β↓βcβ\downarrow β_cβ↓βc​, and α≃0.86α\simeq 0.86α≃0.86 for β↑βcβ\uparrow β_cβ↑βc​, even for small driving. A Floquet analysis shows the nature of the criticality, which is dynamical, with implications that remain unseen and are mostly impossible when the system is in thermal equilibrium.
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.
ABSTRACT. This paper studies a generalization of the Curie-Weiss model (the Ising model on a complete graph) to quantum mechanics. Using a natural probabilistic representation of this model, we give a complete picture of the phase diagram of the model in the parameters of inverse temperature and transverse field strength. Further analysis computes the critical exponent for the decay of the order parameter in the approach to the critical curve and gives useful stability properties of a variational problem associated with the representation. 1.
We study the fate of the Ising model and its universal properties when driven by a rapid periodic drive and weakly coupled to a bath at equilibrium. The far-from-equilibrium steady-state regime is accessed by means of a Floquet mean-field approach. We show that, depending on the details of the bath, the drive can strongly renormalize the critical temperature to higher temperatures, modify the critical exponents, or even change the nature of the phase transition from second to first order after the emergence of a tricritical point. Moreover, by judiciously selecting the frequency of the field and by engineering the spectrum of the bath, one can drive a ferromagnetic Hamiltonian to an antiferromagnetically ordered phase and vice versa.
The Curie–Weiss model is an exactly solvable model of ferromagnetism that allows one to study thermodynamic functions in detail, in particular their properties near the critical temperature. In this model every magnetic moment interacts with every other magnetic moment. Because of its simplicity and the correctness of at least some of its predictions, the Curie–Weiss model occupies an important place in the statistical mechanics literature and its application to information theory. It is frequently presented as an introduction to the Ising model or to spin-glass models, and usually only general features of the Curie–Weiss model are presented. In this paper, we discuss the properties of this model in a rather detailed way. We present the exact, approximate and numerical results for this particular model. The exact expression for the limiting magnetic field is derived.
Phase transitions are typically accompanied by nonanalytic behaviors of the free energy, which can be explained by considering the zeros of the partition function in the complex plane of the control parameter and their approach to the critical value on the real axis as the system size is increased.Recent experiments have shown that partition function zeros are not just a theoretical concept.They can also be determined experimentally by measuring fluctuations of thermodynamic observables in systems of finite size.Motivated by this progress, we investigate here the partition function zeros for the Curie-Weiss model of spontaneous magnetization using our recently established cumulant method.Specifically, we extract the leading Fisher and Lee-Yang zeros of the Curie-Weiss model from the fluctuations of the energy and the magnetization in systems of finite size.We develop a finite-size scaling analysis of the partition function zeros, which is valid for mean-field models and which allows us to extract both the critical values of the control parameters and the critical exponents, even for small systems that are away from criticality.We also show that the Lee-Yang zeros carry important information about the rare magnetic fluctuations as they allow us to predict many essential features of the large-deviation statistics of the magnetization.This finding may constitute a profound connection between Lee-Yang theory and large-deviation statistics.
The critical properties characterizing the formation of the Floquet time crystal in the prethermal phase are investigated analytically in the periodically driven O(N)O(N)O(N) model. In particular, we focus on the critical line separating the trivial phase with period synchronized dynamics and the absence of long-range spatial order from the nontrivial phase where long-range spatial order is accompanied by period-doubling dynamics. In the vicinity of the critical line, with a combination of dimensional expansion and exact solution for $N\ensuremath{\rightarrow}\ensuremath{\infty}$, we determine the exponent $\ensuremath{\nu}$ that characterizes the divergence of the spatial correlation length of the equal-time correlation functions, the exponent $\ensuremath{\beta}$ characterizing the growth of the amplitude of the order parameter, as well as the initial-slip exponent $\ensuremath{\theta}$ of the aging dynamics when a quench is performed from deep in the trivial phase to the critical line. The exponents $\ensuremath{\nu},\ensuremath{\beta},\ensuremath{\theta}$ are found to be identical to those in the absence of the drive. In addition, the functional form of the aging is found to depend on whether the system is probed at times that are small or large compared to the drive period. The spatial structure of the two-point correlation functions, obtained as a linear response to a perturbing potential in the vicinity of the critical line, is found to show algebraic decays that are longer ranged than in the absence of a drive, and besides being period doubled are also found to oscillate in space at the wave vector $\ensuremath{\omega}/(2v), v$ being the velocity of the quasiparticles, and $\ensuremath{\omega}$ being the drive frequency.