The Ising model places spins σᵢ ∈ {±1} on a lattice with Hamiltonian:
Onsager (1944) solved the 2D model exactly: the critical temperature is T_c = 2J/k_B ln(1+√2) ≈ 2.269
At T_c, spontaneous magnetization vanishes continuously, correlation length diverges (ξ → ∞), and the system exhibits scale-free cluster structure. Critical exponents: β=1/8, ν=1, γ=7/4. The left canvas shows spins (bright=+1, dark=−1). The right shows a space-time history of ⟨M⟩. The Wolff algorithm flips entire clusters and dramatically reduces critical slowing down.