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.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.

arXiv:2602.05212Feb 5, 20260 citations

Critical exponents of scaling with a parameter in spin systems

Masuo Suzuki

Physics Letters AMay 1, 197115 citations

A new approach to critical exponents in phase-transitions of spin-systems

G. Prüßner, Damien Loison, K. D. Schotte

Physica A Statistical Mechanics and its ApplicationsOct 1, 20001 citation

Renormalization Group and Critical Exponents for Classical Heisenberg Ferromagnet with Dipole–Dipole Interaction

S. Fakioǧlu

Abstract The renormalization group technique is applied to the three‐dimensional classical Heisenberg spin system with modified dipole–dipole interaction. For this system, the values and dependence of several critical exponents are investigated. These exponents are: α, the specific heat exponent; β and δ, the magnetization exponents; γ, the susceptibility exponent; η, the correlation function exponent; and v, the correlation length exponent. All the critical exponents except β depend upon the potential range σ. When the potential range σ = 3/2, the modified dipole–dipole results agree with mean‐field values of the critical exponents.Abstract The renormalization group technique is applied to the three‐dimensional classical Heisenberg spin system with modified dipole–dipole interaction. For this system, the values and dependence of several critical exponents are investigated. These exponents are: α, the specific heat exponent; β and δ, the magnetization exponents; γ, the susceptibility exponent; η, the correlation function exponent; and v, the correlation length exponent. All the critical exponents except β depend upon the potential range σ. When the potential range σ = 3/2, the modified dipole–dipole results agree with mean‐field values of the critical exponents.

physica status solidi (b)Mar 1, 19808 citations

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.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 ApplicationsJun 8, 20260 citations

Quantum critical phenomena in spin systems

Subir Sachdev A

Jan 1, 19970 citations

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.