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:2602.05212

Mean-field behavior of the finite size Ising model near its critical point

Daniela Olascoaga-Rodríguez, Francisco Sastre, Vı́ctor Romero-Rochı́n

Published Feb 5, 2026• 0 citations
arXivHTMLPDF

Abstract

Universality classes encompass the analogous thermodynamic behavior of unlike physical systems, at different spatial dimensions $d$, in the vicinity of their critical point. Critical exponents define these classes, with the Ising model being the outstanding prototype that elucidates the differences from the mean-field category, believed to be valid above a critical dimension only. Here, in apparent striking contradiction to the Ising universality class, we demonstrate that the critical behavior of a finite Ising system of $N$ spins in $d = 3$ obeys mean-field Landau theory in the vicinity of its critical point, with classical critical exponents. Yet, when expressed in terms of the linear size $L$ of the system, the free energy unveils its proper finite-size scaling form, from which the thermodynamic limit critical temperature $T_c$ and the Ising critical exponents $ν$, $γ$ and $β$ can be identified. We find that the larger the size $L$, the smaller the mean-field region, shrinking to zero in the thermodynamic limit. These conclusions are achieved via the use of an alternative approach to collect data from a Monte Carlo simulation of a three-dimensional Ising model that allows for the evaluation of the free energy per spin $f = f(T,m;L)$ and of the coexistence curve, or spontaneous magnetization at zero magnetic field, $m_{\rm coex} = m(T;L)$ as functions of temperature $T$ and magnetization per spin $m = M/N$. Our results suggest a revision of the role of mean-field theory in the elucidation of critical phenomena.

DOI
10.48550/arxiv.2602.05212

Related papers

Mean-field behavior of the finite size Ising model near its critical point

Daniela Olascoaga-Rodríguez, Francisco Sastre, Vı́ctor Romero-Rochı́n

Universality classes encompass the analogous thermodynamic behavior of unlike physical systems, at different spatial dimensions d , in the vicinity of their critical point. Critical exponents define these classes, with the Ising model being the outstanding prototype that elucidates the differences from the mean-field category, believed to be valid above a critical dimension only. Here, in apparent striking contradiction to the Ising universality class, we demonstrate that the critical behavior of a finite Ising system of N spins in d = 3 obeys mean-field Landau theory in the vicinity of its critical point, with classical critical exponents. Yet, when expressed in terms of the linear size L of the system, the free energy unveils its proper finite-size scaling form, from which the thermodynamic limit critical temperature T c and the Ising critical exponents ν , γ and β can be identified. We find that the larger the size L , the smaller the mean-field region, shrinking to zero in the thermodynamic limit. These conclusions are achieved via the use of an alternative data analysis of Monte Carlo simulations of a three-dimensional Ising model that allows for the evaluation of the free energy per spin f = f ( T , m ; L ) and of the coexistence curve, or spontaneous magnetization at zero magnetic field, m coex = m ( T ; L ) as functions of temperature T and magnetization per spin m = M / N . Our results suggest a revision of the role of mean-field theory in the elucidation of critical phenomena.

Physica A Statistical Mechanics and its Applications•Jun 8, 2026•0 citations

Finite-size scaling in Ising-like systems with quenched random fields: Evidence of hyperscaling violation

Richard L. C. Vink, Timo Fischer, Kurt Binder

In systems belonging to the universality class of the random field Ising model, the standard hyperscaling relation between critical exponents does not hold, but is replaced with a modified hyperscaling relation. As a result, standard formulations of finite-size scaling near critical points break down. In this work, the consequences of modified hyperscaling are analyzed in detail. The most striking outcome is that the free-energy cost ΔF of interface formation at the critical point is no longer a universal constant, but instead increases as a power law with system size, ΔF∝L(θ), with θ as the violation of hyperscaling critical exponent and L as the linear extension of the system. This modified behavior facilitates a number of numerical approaches that can be used to locate critical points in random field systems from finite-size simulation data. We test and confirm the approaches on two random field systems in three dimensions, namely, the random field Ising model and the demixing transition in the Widom-Rowlinson fluid with quenched obstacles.

arXiv:1008.3299•Physical Review E•Nov 30, 2010•44 citations

Computer simulation of the critical behavior of 3D disordered ising model

V. V. Prudnikov, Pavel Vladimirovich Prudnikov, A. N. Vakilov +1 more

The critical behavior of the disordered ferromagnetic Ising model is studied numerically by the Monte Carlo method in a wide range of variation of concentration of nonmagnetic impurity atoms. The temperature dependences of correlation length and magnetic susceptibility are determined for samples with various spin concentrations and various linear sizes. The finite-size scaling technique is used for obtaining scaling functions for these quantities, which exhibit a universal behavior in the critical region; the critical temperatures and static critical exponents are also determined using scaling corrections. On the basis of variation of the scaling functions and values of critical exponents upon a change in the concentration, the conclusion is drawn concerning the existence of two universal classes of the critical behavior of the diluted Ising model with different characteristics for weakly and strongly disordered systems.

arXiv:0709.1450•Journal of Experimental and Theoretical Physics•Aug 1, 2007•35 citations

Anomalous mean-field behavior of the fully connected Ising model

Louis Colonna-Romano, Harvey Gould, W. Klein

Although the fully connected Ising model does not have a length scale, we show that the critical exponents for thermodynamic quantities such as the mean magnetization and the susceptibility can be obtained using finite size scaling with the scaling variable equal to N, the number of spins. Surprisingly, the mean value and the most probable value of the magnetization are found to scale differently with N at the critical temperature of the infinite system, and the magnetization probability distribution is not a Gaussian, even for large N. Similar results inconsistent with the usual understanding of mean-field theory are found at the spinodal. We relate these results to the breakdown of hyperscaling and show that hyperscaling can be restored by increasing N while holding the Ginzburg parameter rather than the temperature fixed, or by doing finite size scaling at the pseudocritical temperature where the susceptibility is a maximum for a given value of N. We conclude that finite size scaling for the fully connected Ising model yields different results depending on how the mean-field limit is approached.

arXiv:1403.4854•Physical Review E•Oct 6, 2014•17 citations

Critical fluctuations at a finite-time dynamical phase transition

Nalina Vadakkayil, Massimiliano Esposito, Jan Meibohm

We explore the critical properties of the recently discovered finite-time dynamical phase transition in the nonequilibrium relaxation of Ising magnets after a temperature quench. The transition is characterized by a sudden switch in the relaxation dynamics and it occurs at a sharp critical time. While previous works have focused either on mean-field interactions or on investigating the critical time, we analyze the critical fluctuations at the phase transition in the nearest-neighbor Ising model on a square lattice using Monte Carlo simulations. By means of a finite-size scaling analysis, we extract the critical exponents for the transition. In two spatial dimensions, we find that the exponents are consistent with those of the two-dimensional Ising universality class when the system is initially in the vicinity of the critical point. For initial temperatures below the critical one, however, the critical exponents differ from the Ising-exponents, indicating a distinct dynamical critical phenomenon.

arXiv:2403.10505•Physical review. E•Dec 30, 2024•6 citations

A nonequilibrium system on a restricted scale-free network

Rafael Alves Dumer, Marta Arantes Godoy

The nonequilibrium Ising model on a restricted scale-free network has been studied with one- and two-spin flip competing dynamics employing Monte Carlo simulations. The dynamics present in the system can be defined by the probability $q$ in which the one-spin flip process simulate the contact with a heat bath at a given temperature $T$, and with a probability ($1-q$) the two-spin flip process mimics the system subjected to an external flux of energy into it. The system network is described by a power-law degree distribution in the form $P(k)\sim k^{-α}$, and the restriction is made by fixing the maximum, $k_{m}$, and minimum, $k_{0}$, degree on distribution for the whole network size. This restriction keeps finite the second and fourth moment of degree distribution, allowing us to obtain a finite critical point for any value of $α$. For these critical points, we have calculated the thermodynamic quantities of the system, such as, the total ${m}_{N}^{F}$ and staggered ${m}_{N}^{AF}$ magnetizations per spin, susceptibility $χ_{N}$, and reduced fourth-order Binder cumulant ${U}_{N}$, for several values of lattice size $N$ and exponent $1\leα\le5$. Therefore, the phase diagram was built and a self-organization phenomena is observed from the transitions between antiferromagnetic AF to paramagnetic P, and P to ferromagnetic F phases. Using the finite-size scaling theory, we also obtained the critical exponents for the system, and a mean-field critical behavior is observed, exhibiting the same universality class of the system on the equilibrium and out of it.

arXiv:2306.04780•Jun 7, 2023•0 citations
Universality classes encompass the analogous thermodynamic behavior of unlike physical systems, at different spatial dimensions ddd, in the vicinity of their critical point. Critical exponents define these classes, with the Ising model being the outstanding prototype that elucidates the differences from the mean-field category, believed to be valid above a critical dimension only. Here, in apparent striking contradiction to the Ising universality class, we demonstrate that the critical behavior of a finite Ising system of NNN spins in d=3d = 3d=3 obeys mean-field Landau theory in the vicinity of its critical point, with classical critical exponents. Yet, when expressed in terms of the linear size LLL of the system, the free energy unveils its proper finite-size scaling form, from which the thermodynamic limit critical temperature TcT_cTc​ and the Ising critical exponents ννν, γγγ and βββ can be identified. We find that the larger the size LLL, the smaller the mean-field region, shrinking to zero in the thermodynamic limit. These conclusions are achieved via the use of an alternative approach to collect data from a Monte Carlo simulation of a three-dimensional Ising model that allows for the evaluation of the free energy per spin f=f(T,m;L)f = f(T,m;L)f=f(T,m;L) and of the coexistence curve, or spontaneous magnetization at zero magnetic field, mcoex=m(T;L)m_{\rm coex} = m(T;L)mcoex​=m(T;L) as functions of temperature TTT and magnetization per spin m=M/Nm = M/Nm=M/N. Our results suggest a revision of the role of mean-field theory in the elucidation of critical phenomena.
Universality classes encompass the analogous thermodynamic behavior of unlike physical systems, at different spatial dimensions d , in the vicinity of their critical point. Critical exponents define these classes, with the Ising model being the outstanding prototype that elucidates the differences from the mean-field category, believed to be valid above a critical dimension only. Here, in apparent striking contradiction to the Ising universality class, we demonstrate that the critical behavior of a finite Ising system of N spins in d = 3 obeys mean-field Landau theory in the vicinity of its critical point, with classical critical exponents. Yet, when expressed in terms of the linear size L of the system, the free energy unveils its proper finite-size scaling form, from which the thermodynamic limit critical temperature T c and the Ising critical exponents ν , γ and β can be identified. We find that the larger the size L , the smaller the mean-field region, shrinking to zero in the thermodynamic limit. These conclusions are achieved via the use of an alternative data analysis of Monte Carlo simulations of a three-dimensional Ising model that allows for the evaluation of the free energy per spin f = f ( T , m ; L ) and of the coexistence curve, or spontaneous magnetization at zero magnetic field, m coex = m ( T ; L ) as functions of temperature T and magnetization per spin m = M / N . Our results suggest a revision of the role of mean-field theory in the elucidation of critical phenomena.
In systems belonging to the universality class of the random field Ising model, the standard hyperscaling relation between critical exponents does not hold, but is replaced with a modified hyperscaling relation. As a result, standard formulations of finite-size scaling near critical points break down. In this work, the consequences of modified hyperscaling are analyzed in detail. The most striking outcome is that the free-energy cost ΔF of interface formation at the critical point is no longer a universal constant, but instead increases as a power law with system size, ΔF∝L(θ), with θ as the violation of hyperscaling critical exponent and L as the linear extension of the system. This modified behavior facilitates a number of numerical approaches that can be used to locate critical points in random field systems from finite-size simulation data. We test and confirm the approaches on two random field systems in three dimensions, namely, the random field Ising model and the demixing transition in the Widom-Rowlinson fluid with quenched obstacles.
The critical behavior of the disordered ferromagnetic Ising model is studied numerically by the Monte Carlo method in a wide range of variation of concentration of nonmagnetic impurity atoms. The temperature dependences of correlation length and magnetic susceptibility are determined for samples with various spin concentrations and various linear sizes. The finite-size scaling technique is used for obtaining scaling functions for these quantities, which exhibit a universal behavior in the critical region; the critical temperatures and static critical exponents are also determined using scaling corrections. On the basis of variation of the scaling functions and values of critical exponents upon a change in the concentration, the conclusion is drawn concerning the existence of two universal classes of the critical behavior of the diluted Ising model with different characteristics for weakly and strongly disordered systems.
Although the fully connected Ising model does not have a length scale, we show that the critical exponents for thermodynamic quantities such as the mean magnetization and the susceptibility can be obtained using finite size scaling with the scaling variable equal to N, the number of spins. Surprisingly, the mean value and the most probable value of the magnetization are found to scale differently with N at the critical temperature of the infinite system, and the magnetization probability distribution is not a Gaussian, even for large N. Similar results inconsistent with the usual understanding of mean-field theory are found at the spinodal. We relate these results to the breakdown of hyperscaling and show that hyperscaling can be restored by increasing N while holding the Ginzburg parameter rather than the temperature fixed, or by doing finite size scaling at the pseudocritical temperature where the susceptibility is a maximum for a given value of N. We conclude that finite size scaling for the fully connected Ising model yields different results depending on how the mean-field limit is approached.
We explore the critical properties of the recently discovered finite-time dynamical phase transition in the nonequilibrium relaxation of Ising magnets after a temperature quench. The transition is characterized by a sudden switch in the relaxation dynamics and it occurs at a sharp critical time. While previous works have focused either on mean-field interactions or on investigating the critical time, we analyze the critical fluctuations at the phase transition in the nearest-neighbor Ising model on a square lattice using Monte Carlo simulations. By means of a finite-size scaling analysis, we extract the critical exponents for the transition. In two spatial dimensions, we find that the exponents are consistent with those of the two-dimensional Ising universality class when the system is initially in the vicinity of the critical point. For initial temperatures below the critical one, however, the critical exponents differ from the Ising-exponents, indicating a distinct dynamical critical phenomenon.
The nonequilibrium Ising model on a restricted scale-free network has been studied with one- and two-spin flip competing dynamics employing Monte Carlo simulations. The dynamics present in the system can be defined by the probability qqq in which the one-spin flip process simulate the contact with a heat bath at a given temperature TTT, and with a probability (1−q1-q1−q) the two-spin flip process mimics the system subjected to an external flux of energy into it. The system network is described by a power-law degree distribution in the form P(k)∼k−αP(k)\sim k^{-α}P(k)∼k−α, and the restriction is made by fixing the maximum, kmk_{m}km​, and minimum, k0k_{0}k0​, degree on distribution for the whole network size. This restriction keeps finite the second and fourth moment of degree distribution, allowing us to obtain a finite critical point for any value of ααα. For these critical points, we have calculated the thermodynamic quantities of the system, such as, the total mNF{m}_{N}^{F}mNF​ and staggered mNAF{m}_{N}^{AF}mNAF​ magnetizations per spin, susceptibility χNχ_{N}χN​, and reduced fourth-order Binder cumulant UN{U}_{N}UN​, for several values of lattice size NNN and exponent 1≤α≤51\leα\le51≤α≤5. Therefore, the phase diagram was built and a self-organization phenomena is observed from the transitions between antiferromagnetic AF to paramagnetic P, and P to ferromagnetic F phases. Using the finite-size scaling theory, we also obtained the critical exponents for the system, and a mean-field critical behavior is observed, exhibiting the same universality class of the system on the equilibrium and out of it.