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

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.

Figures (7)

• Figure 1: Finite size free energy $f(m,T;L)$ as a function of magnetization $m$, above, at, and below $T_c^L$, for $L = 22$. These curves are not fittings, appearing continuous due to the large number of magnetization points and the number of configurations, ${\cal N} \approx 5\times 10^{13}$ for each temperature
• Figure 2: Fitting coefficients $A_2(T,L)$ and $A_4(T,L)$ for $L = 22$, in the vicinity of the critical point $T_c^L$, see (\ref{['ffit']}). One finds, very accurately, $A_2(T,L) \approx a_0(L) (T - T_c^L)$, thus fitting $a_0(L)$ and $T_c^L$, the latter shown with a vertical dashed line in both plots. Then, we fit $A_4(T,L) \approx b(L) + b^\prime(L)(T-T_c^L)$.
• Figure 3: Non linear fittings of $T_c^L \approx T_c + {\cal L}_0 L^{-1/\nu}$ vs $L^{-1}$. The dots are the results of the fittings for $L = 10, 14, 22, 26, 30, 50$ given in the Appendix.
• Figure 4: Non linear fittings of Landau coefficients $a_0(L) = {\cal A} L^{(\gamma-1)/\nu}$ and $b(L) = \mathcal{B} L^{(\gamma-2\beta)/\nu}$, shown with the orange curves. The dots are the results of the fittings for $L = 10, 14, 22, 26, 30, 50$ given in SM.
• Figure 5: Coexistence curves, or spontaneous magnetization, $T$ vs $m$, at $H = 0$, for systems with our studied values of $L$. We also include values from standard MonteCarlo simulations marked as MC, for $T \le 4$, where all points coalesce to the thermodynamic limit curve. The inset shows the spontaneous magnetizations near their critical points, each with an even polynomial fitting. The solid black curve is an empirical fitting of the coexistence curve in the thermodynamic limit. See text