The Ising model and the mean-field approximation

Take Ising spins s_i = ±1 with energy H = −JΣ⟨ij⟩s_i s_j − hΣ_i s_i. Count each nearest-neighbour pair once. J > 0 favours alignment; h is the field energy per spin. Each site has z neighbours. Define m = ⟨s_i⟩. Mean field replaces each neighbouring spin by m, giving local field energy zJm+h. The full model is not always intractable: the one-dimensional model and the zero-field square-lattice model have exact solutions.

For a mean-field spin, set x = (zJm+h)/(k_B T), with Boltzmann's constant k_B. The weights for s = +1 and −1 are exp(x) and exp(−x). Divide each weight by their sum to obtain probabilities. Then m = (+1)P(+) + (−1)P(−) = (exp(x)−exp(−x))/(exp(x)+exp(−x)) = tanh(x). Thus m = tanh((zJm+h)/(k_B T)). Graph y=m and y=tanh((zJm+h)/(k_B T)); intersections are self-consistent averages.

At h = 0, set a = zJ/(k_B T), g(m) = tanh(am). Under m → g(m), a small perturbation δ near a root becomes approximately g'(m)δ: |g'| < 1 shrinks it; |g'| > 1 grows it. At zero, g' = a, so a = 1 gives T_c = zJ/k_B. Above T_c only zero is a root. Below T_c, g(m) > m for small m > 0 but g(1) < 1, giving roots ±m* by continuity and odd symmetry. The only roots are 0 and ±m*: zero is unstable; ±m* are stable since 0 < a(1−m*²) < 1.

Mean field replaces neighbour fluctuations by their average, so it neglects correlations between different spins. This can even change whether a transition exists. The infinite one-dimensional nearest-neighbour Ising chain at zero field has no finite-temperature ordered phase, while mean field predicts T_c = 2J/k_B because z = 2.

Near T_c at h = 0, set a = T_c/T. For small am, tanh(am) ≈ am − (am)^3/3; higher powers are negligible to this order. Substitute into m = tanh(am) and divide by nonzero m to get m² ≈ 3(a−1)/a³. Since a−1 ≈ τ = (T_c−T)/T_c > 0 and a ≈ 1 below T_c, the leading law is m ∝ τ^(1/2), with β = 1/2. Reducing τ by a factor of nine reduces m by a factor of three in this approximation.

For short-range Ising systems, correlated fluctuations generally change critical exponents below the upper critical dimension d_c = 4. Above four dimensions the leading exponents take mean-field values; at four dimensions logarithmic corrections can occur. Away from criticality, mean field may be useful, but it is not generally exact.