Replace the fluctuating neighbors of each spin by their average, and the Ising model becomes a system of independent spins in an effective field. The resulting self-consistency equation has a phase transition at Tc = Jz/kB, with critical exponents α = 0, β = 1/2, γ = 1 and δ = 3. Landau theory recovers them all from a fourth-order polynomial in the order parameter.
Based on lecture notes by Sang Hoon Lee for Chapter 2, following K. Christensen and N. R. Moloney, Complexity and Criticality (2005). Units: kB = 1 and J = 1, so Tc = z. References
Each spin interacts only with its z nearest neighbors, and those neighbors fluctuate. Mean-field theory replaces them by their average magnetization m. What is left is one spin in a steady effective field, which must reproduce the m it was built from.
Write each spin as its average plus a fluctuation, si = ⟨si⟩ + (si − ⟨si⟩), with ⟨si⟩ = m. The interaction energy then contains a term of second order in the fluctuations, (si − m)(sj − m), which mean-field theory neglects. Each of the Nz/2 bonds is then sisj ≈ (si + sj)m − m², and summing gives
This is a system of non-interacting spins in an effective field Jzm + H, the external field plus an internal field from the z neighbors, plus a constant. Its partition function factorizes as before:
The order parameter m is not a free variable: in equilibrium it takes the value that minimizes f. Setting ∂f/∂m = 0, or equivalently m = −∂f/∂H, gives the self-consistency equation m = tanh(βJzm + βH). At H = 0 it reads m0 = tanh[(Tc/T)m0] with Tc = Jz/kB. The slope of the right-hand side at m = 0 is Tc/T, so nonzero solutions exist only below Tc.
| Lattice | z | Mean-field kBTc/J | Exact or best known kBTc/J |
|---|---|---|---|
| Chain (d = 1) | 2 | 2 | 0 (no transition) |
| Square (d = 2) | 4 | 4 | 2.269 (Onsager) |
| Simple cubic (d = 3) | 6 | 6 | ≈ 4.51 (numerical) |
Near Tc the self-consistency equation can be expanded, tanh x ≈ x − x³/3, giving m0 ≈ ±√3 (T/Tc)3/2 [(Tc − T)/T]1/2, so β = 1/2. At T = Tc the same expansion gives m(Tc, H) ≈ sign(H)(3βc|H|)1/3, so δ = 3. Differentiating the self-consistency equation gives the susceptibility
so γ = 1 on both sides, with the universal amplitude ratio Γ+/Γ− = 2. The energy is ε(T, 0) = −(Jz/2)m0², zero above Tc, so the specific heat jumps from 3kB/2 to 0 at Tc: a discontinuity rather than a divergence, α = 0.
The mean-field exponents all come from Taylor expansions around Tc, where m is small. Landau’s approach expands the free energy itself in powers of the order parameter. With ln cosh x = x²/2 − x⁴/12 + …, the mean-field f becomes
f0 is the entropy of 2N configurations. The coefficient of m is set by the field, the m² coefficient changes sign at Tc, and the m⁴ coefficient stays positive. Minimizing, −H + 2a2(T − Tc)m + 4a4m³ = 0, gives m0 = ±√a2(Tc − T)/(2a4) = ±√3/Tc (Tc − T)1/2, χ = 1/[2a2(T − Tc) + 12a4m0²], m(Tc, H) = sign(H)(3βc|H|)1/3, and f = f0 − (3/4Tc)(T − Tc)² below Tc, so c = −T ∂²f/∂T² jumps to 3kB/2. Every exponent is recovered, even though the expansion loses the details far from Tc.
The sketch below shows f − f0 = −Hm + a2(T − Tc)m² + a4m⁴ for H < 0, H = 0 and H > 0 (rows) and T < Tc, T = Tc and T > Tc (columns), with a ball at the global minimum. Down the left column the ball jumps from one well to the other: a first-order transition at H = 0. Along the middle row the two wells merge continuously: a second-order transition at Tc.
If the free energy is analytic near the critical point, it can be expanded in a general order parameter φ, which is defined implicitly by minimizing f, so it is not an independent variable like T and H: f(T, H; φ) = Σk αk(T, H)φk. At H = 0 the up–down symmetry f(φ) = f(−φ) forbids odd powers, and the simplest form describing a continuous transition is a fourth-order polynomial. Expanding the coefficients to leading order around (Tc, 0):
α̃0 is the entropic part, α̃1 = −1, α̃2 > 0 so the φ² coefficient changes sign at Tc, and α̃4 > 0 keeps f bounded. Any system described this way has α = 0, β = 1/2, γ = 1 and δ = 3. Try the coefficients:
Mean-field theory predicts a second-order transition at a finite Tc in zero field, which is correct for d > 1, and a line of first-order transitions at H = 0 for T < Tc ending at the critical point. Its Tc grows with J and z, which is qualitatively right, and like pc in percolation Tc depends on lattice details. Its exponents and the ratio Γ+/Γ− = 2 do not depend on J or z, so they are universal. But by neglecting fluctuations it predicts the same exponents in every dimension, and a transition even in d = 1. The exact values differ below four dimensions:
| Exponent | Definition | Mean field / Landau | 2D Ising (exact) | 3D Ising (numerical) |
|---|---|---|---|---|
| α | c ∝ |T − Tc|−α | 0 (jump) | 0 (logarithmic) | ≈ 0.110 |
| β | m0 ∝ (Tc − T)β | 1/2 | 1/8 | ≈ 0.326 |
| γ | χ ∝ |T − Tc|−γ | 1 | 7/4 | ≈ 1.237 |
| δ | m(Tc, H) ∝ |H|1/δ | 3 | 15 | ≈ 4.79 |
Key reference. K. Christensen and N. R. Moloney, Complexity and Criticality, Imperial College Press, London (2005), Chapter 2.