Self-consistency: m = tanh(βJzm + βh) — phase transition at T_c = Jz/k_B
Self-consistency diagram & free energy landscape
Magnetization vs Temperature (phase diagram)
Stable magnetization m*: —
T_c = Jz/k_B: 6.00 J/k_B
Free energy F(m*): —
Mean field theory
Each spin sees effective field h_eff = Jzm + h
m = tanh(β·h_eff) T > T_c: unique solution m=0 (paramagnetic) T < T_c: two symmetric solutions ±m* (ferromagnetic) Landau expansion: F ≈ a(T−T_c)m² + bm⁴ Critical exponent: β=½ (MF), exact=⅛ (2D)
Hysteresis
With h≠0: sweep field to see hysteresis loop.
Spontaneous magnetization breaks Z₂ symmetry.
Order parameter: m — continuous at T_c (2nd order).