Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1301.2141 · European Journal of Physics

Curie–Weiss magnet—a simple model of phase transition

M Kochmański, T Paszkiewicz, S. Wolski

Published Oct 22, 2013• 86 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/0143-0807/34/6/1555

Related papers

Ferromagnetism

Lawrence S. Schulman

Abstract Magnetism and order parameters are discussed in both mean field theory and exact treatments. An example of stochastic dynamics for the Ising model is given. The Curie–Weiss model is defined as a system in which mean field theory is exact. For this system (consisting of N spins) the fluctuations at the critical point are the fourth root of N, rather than the square root as is the case when the variables are independent.

•Jan 18, 2022•0 citations

Introduction to critical phenomena

Michel Le Bellac, G. Barton

Abstract This first chapter is an elementary introduction to the models and methods that will be used later. The ferromagnetic (Curie) transition serves as our prototype for all second-order phase transitions, and serves also to introduce the notions of broken symmetry and of order parameter. The basic model of ferromagnetism, namely the Ising model, is described in Section 1.2, and solved for an elementary case. The reason why this model is so important is that it has an exact solution in two dimensions, which exhibits spontaneous magnetization. Section 1.3 is devoted to a first approach to mean-field methods, which remain amongst the most frequently exploited approximations even today. They will lead us to define a first set of critical exponents; the listing of such exponents will then be completed in Section 1.4, once correlation functions have been defined. Finally, in Section 1.5 a qualitative description of critical phenomena will enable us, in a first approximation, to define the concept of scale invariance and its relation to the behaviour of the theory under dilatations.

•Mar 26, 1992•5 citations

Critical behavior of the driven Curie-Weiss model

Ruohan Xu, Faezeh Khodabandehlou, Christian Maes

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.

arXiv:2607.05130•Jul 6, 2026•0 citations

Lee-Yang theory of the Curie-Weiss model and its rare fluctuations

Aydin Deger, Christian Flindt

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.

arXiv:2002.01269•Physical Review Research•Jul 2, 2020•28 citations

Thermodynamic properties of the itinerant-boson ferromagnet

Chengjun Tao, Peilin Wang, Jihong Qin +1 more

Thermodynamics of a spin-1 Bose gas with ferromagnetic interactions is investigated via the mean-field theory. It is apparently shown in the specific-heat curve that the system undergoes two phase transitions, the ferromagnetic transition and Bose-Einstein condensation, with the Curie point above the condensation temperature. Above the Curie point, the susceptibility fits the Curie-Weiss law perfectly. At a fixed temperature, the reciprocal susceptibility is also in a good linear relationship with the ferromagnetic interaction.

arXiv:0810.1094•Physical Review B•Oct 3, 2008•12 citations

Generalization of the Curie-Weiss Model to the D-dimensional Spin System

Hyung-june Woo, Eun Kyung Lee, Eok-Kyun Lee

The critical behavior of the classical D-dimensional spin model (D2), which is intermediate model that link up the Ising (D = 1) and the spherical model (D = ), is studied for the case of constant coupling interaction independent of the spin-spin distance (Curie-Weiss model). Analytical results show that the critical behavior of the present model is in quantitative agreement with the prediction of the phenomenological mean-field theory independent of D. Critical temperature is calculated to be T=k/JD. This gives a quantitative explanation of the relationship between the spin degree of freedom and the critical temperature.

Bulletin of the Korean Chemical Society•Aug 1, 1993•0 citations
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.
Abstract This first chapter is an elementary introduction to the models and methods that will be used later. The ferromagnetic (Curie) transition serves as our prototype for all second-order phase transitions, and serves also to introduce the notions of broken symmetry and of order parameter. The basic model of ferromagnetism, namely the Ising model, is described in Section 1.2, and solved for an elementary case. The reason why this model is so important is that it has an exact solution in two dimensions, which exhibits spontaneous magnetization. Section 1.3 is devoted to a first approach to mean-field methods, which remain amongst the most frequently exploited approximations even today. They will lead us to define a first set of critical exponents; the listing of such exponents will then be completed in Section 1.4, once correlation functions have been defined. Finally, in Section 1.5 a qualitative description of critical phenomena will enable us, in a first approximation, to define the concept of scale invariance and its relation to the behaviour of the theory under dilatations.
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.
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.
Thermodynamics of a spin-1 Bose gas with ferromagnetic interactions is investigated via the mean-field theory. It is apparently shown in the specific-heat curve that the system undergoes two phase transitions, the ferromagnetic transition and Bose-Einstein condensation, with the Curie point above the condensation temperature. Above the Curie point, the susceptibility fits the Curie-Weiss law perfectly. At a fixed temperature, the reciprocal susceptibility is also in a good linear relationship with the ferromagnetic interaction.
The critical behavior of the classical D-dimensional spin model (D2), which is intermediate model that link up the Ising (D = 1) and the spherical model (D = ), is studied for the case of constant coupling interaction independent of the spin-spin distance (Curie-Weiss model). Analytical results show that the critical behavior of the present model is in quantitative agreement with the prediction of the phenomenological mean-field theory independent of D. Critical temperature is calculated to be T=k/JD. This gives a quantitative explanation of the relationship between the spin degree of freedom and the critical temperature.