Mean-Field Ising Model

Self-consistency equation · spontaneous magnetization · phase diagram

T = 2.00
h = 0.00
z = 6 (3D cubic)
Self-consistency: m = tanh(βJzm + βh)
Magnetization m vs T
Mean-field Ising: replace all neighbors with average field → self-consistency equation m = tanh(βJzm + βh), where β=1/T, z = coordination number.
Phase transition at T_c = Jz/k_B (β_c·Jz=1): for T < T_c and h=0, spontaneous magnetization m≠0 exists (two solutions ±m*).
Symmetry breaking: choosing ±m* = choosing a ground state. The graphical intersection of y=tanh(βJzm) and y=m reveals stable/unstable fixed points. In 3D: T_c ≈ 4.5 K for iron (mean field overestimates by ~2×).