The Ising model and mean-field theory

Place variables $s_i=\pm1$ on a lattice and assign ferromagnetic coupling $J>0$ to nearest-neighbor pairs. Count each unordered bond once. A field term $-h\sum_i s_i$ biases the global orientation. On a regular bulk lattice with coordination number $z$, there are approximately $Nz/2$ bonds; boundaries modify that count for a finite open sample. Energy favors alignment, while the exponentially many disordered configurations can win entropically at high temperature.

Unlike independent spins, the partition function no longer factorizes into one-site factors because neighboring spins appear together in the energy. Dimensionality, coupling range, and boundary conditions matter. A finite zero-field system with spin-flip symmetric boundaries has exactly zero mean total magnetization when averaged over the complete equilibrium ensemble, even if most configurations have a large positive or negative magnetization. A symmetry-broken thermodynamic phase requires a specified limiting or conditioning procedure rather than discarding the opposite sign by assertion.

Write a spin as its mean plus a fluctuation, and neglect products of the neighboring fluctuations. Each spin then sees an effective field $h+zJm$. The one-spin thermal mean in that field must equal the mean $m$ used to construct it. This produces a self-consistency equation, not an externally prescribed magnetization. The decoupling also produces a constant correction that prevents double-counting interaction energy when constructing the mean-field free energy.

At zero field, expand the hyperbolic tangent near zero magnetization. The disordered solution changes its mean-field stability when $\beta zJ=1$, giving $T_c^{\mathrm{MF}}=zJ/k_B$. Above that value, linearizing in a small field yields susceptibility per spin $1/(k_BT-zJ)$. The divergence belongs to the infinite-system approximation. A numerical iteration of the self-consistency map can locate solutions, but its iteration index is not physical time and its convergence should not be interpreted as a measured magnetic relaxation rate.

Below the mean-field transition, two oppositely magnetized solutions appear at zero field. Expanding the mean-field free energy near the transition gives a positive quartic term and predicts a square-root onset of spontaneous magnetization. Choosing a phase means first taking a thermodynamic limit with a small selecting field, then allowing that field to vanish. Reversing those limits restores the exact zero finite-system mean. Absolute magnetization and long-distance correlations can still reveal ordering tendencies in finite calculations.

Mean field discards spatial correlations, which become especially important near criticality and in low dimensions. A one-dimensional short-range Ising chain has no positive-temperature ordered phase, despite the nonzero transition temperature predicted by substituting $z=2$ into the mean-field formula. Two- and three-dimensional critical exponents also differ from the simplest mean-field values. Large coordination, long-range couplings with suitable scaling, or dimensions above an appropriate upper critical dimension can improve its usefulness, but none of those conditions should be assumed for an arbitrary lattice.