Ising Model: Exact Mean-Field Solution

The self-consistency equation m = tanh(βJzm + βh) is solved graphically. Where the curve intersects y=x gives the stable magnetizations. At Tc = Jz/k a continuous phase transition occurs.

Self-Consistency
Phase Diagram
Free Energy

Parameters

m = tanh(βJzm + βh)
T_c = Jz / k_B
β = 1 / (k_B T)

Solution

Magnetization m*---
T_c = Jz/k---
Reduced t = T/T_c---
m ~ (1-T/Tc)^ββ=0.5
Phase---
Critical exponents (MF):
β = 1/2 (magnetization)
γ = 1 (susceptibility)
δ = 3 (m vs h at T_c)
ν = 1/2 (correlation length)
Exact (2D): β=1/8, γ=7/4