2D Ising model + mean-field theory: ferromagnet to paramagnet at T_c = 2J/k ln(1+√2) ≈ 2.269
2.00
0.00
10
T: 2.00
Magnetization M: 0
Energy/site: 0
Susceptibility χ: —
Ising Model: spins σᵢ ∈ {±1} on a 2D square lattice. H = −J Σ σᵢσⱼ − h Σ σᵢ. Simulated with Metropolis algorithm: flip spin i if ΔE < 0, else flip with probability e^(−ΔE/kT).
Phase transition at T_c ≈ 2.269 J/k (Onsager 1944 exact solution — first exact result in statistical mechanics). Below T_c, spontaneous magnetization M ≠ 0 (ferromagnet). Above T_c, M = 0 (paramagnet). Near T_c: M ~ (T_c − T)^β with β = 1/8, χ ~ |T − T_c|^−γ with γ = 7/4 (2D Ising critical exponents).
Mean-field theory predicts T_c = zJ/k = 4J/k (z=4 for square lattice), β = 1/2 — wrong exponents because it ignores fluctuations. The right panel shows the M-T phase diagram with both simulation data and mean-field prediction.