Mean-field behavior of the finite size Ising model near its critical point
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 $\nu$, $\gamma$ and $\beta$ 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.