A grid of magnetic spins, each pointing up or down. Each spin "wants" to align with its neighbors (energy lowered by alignment). Thermal noise randomizes them. Below the Curie temperature T_c ≈ 2.269 J/k_B (Onsager 1944), the system spontaneously magnetizes — billions of spins align. Above T_c, randomness wins. The simplest non-trivial phase transition; the cornerstone of statistical physics.
why a critical temperature is unavoidable
Each spin σ_i ∈ {+1, -1}. Energy:
E = -J Σ_{⟨i,j⟩} σ_i σ_j - h Σ_i σ_i
(J > 0 favors alignment; h is external magnetic field.)
Boltzmann probability: P(state) ∝ e^(-E/kT). At low T, the system finds low-energy (aligned) configurations. At high T, all configurations equally likely → random.
Onsager's 1944 exact solution (a tour-de-force of mathematical physics) gave the critical temperature for the 2D Ising model:
k_B T_c / J = 2 / ln(1 + √2) ≈ 2.269
At this exact temperature, the magnetization develops critical exponents:
• Magnetization M ~ (T_c - T)^β with β = 1/8
• Susceptibility χ ~ |T - T_c|^(-γ) with γ = 7/4
• Correlation length ξ ~ |T - T_c|^(-ν) with ν = 1
Universality: these exponents don't depend on the lattice details. ANY 2D system with the same symmetries (Z2) has these exact exponents. Water vapor at the critical point, binary alloys, lipid bilayers — same exponents. This is the deepest fact about phase transitions: most microscopic details are irrelevant. The renormalization group (Wilson Nobel 1982) explains why.
Try it: at T = 2.27 (right at critical), watch domains of all sizes form and dissolve. At T = 1.5 (cold): the system locks into one ordered state. At T = 4 (hot): pure noise. The crossover is sharp, with the longest correlations appearing right at T_c.