Mean-field behavior of the finite size Ising model near its critical point
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910012228501504 |
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| author | Olascoaga-Rodríguez, D. Sastre, F. Romero-Rochín, V. |
| author_facet | Olascoaga-Rodríguez, D. Sastre, F. Romero-Rochín, V. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_05212 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Mean-field behavior of the finite size Ising model near its critical point Olascoaga-Rodríguez, D. Sastre, F. Romero-Rochín, V. Statistical Mechanics 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. |
| title | Mean-field behavior of the finite size Ising model near its critical point |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2602.05212 |