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 Tc 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, mcoex=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.