A lattice of magnetic spins flips between up and down states. Above the critical temperature, spins are random (paramagnetic). Below it, they align spontaneously (ferromagnetic) — a phase transition emerges.
Metropolis-Hastings algorithm: each spin considers flipping based on its neighbours' alignment and temperature. The partition function determines equilibrium properties.
Adjust temperature across the critical point (T_c ≈ 2.269 for 2D). Watch domains form below T_c and disorder appear above. Observe the magnetisation curve.
Lars Onsager solved the 2D Ising model exactly in 1944 — one of the greatest achievements in statistical mechanics. The 3D Ising model remains unsolved analytically after 100+ years.