Upper Critical Dimension

Controls

Dimension d 3.0 Ginzburg parameter G 0.50 Temperature t = (T-Tc)/Tc 0.10

MF valid

Fluctuations

Ginzburg Criterion

Gi = (G/ξ^d) / ⟨φ²⟩²

Gi ~ |t|^ν(4−d) → d_uc=4

Ginzburg (1960): MF breaks down when fluctuations ∼ MF order param:

Gi < 1: MF is valid

Gi > 1: fluctuations dominate

For d > 4: Gi → 0 near Tc → MF is asymptotically exact. For d < 4: Gi → ∞ near Tc → full RG treatment needed. d_uc = 4 for φ&sup4; theory (Ising universality).

d_uc = 6 (tricritical, percolation)

d_uc = 8 (cubic anisotropy)