Ising
Phase transitions in a 2D lattice
Ising (1925) · Wolff (1989) · Tsarev et al. (2019)
The 2D Ising model is the smallest system that exhibits a phase transition with every expected observable: an order parameter, a diverging susceptibility, a specific-heat singularity, a universal Binder cumulant at criticality. Metropolis-Hastings gives you the honest sluggishness of single-spin dynamics; Wolff defeats critical slowing down by flipping whole correlated clusters at once. Onsager (1944) solved it exactly. We sample from it.
α = 0 (log) · η = 1/4 · δ = 15
U* ≈ 0.6107 (Binder cumulant crossing)
χ_max ∼ L^(γ/ν) = L^(7/4)
Set T a hair above T_c and switch to the cluster view. You'll see the percolating cluster — the one that touches opposite edges — struggle to maintain itself. The fractal dimension of that cluster at criticality is exactly d_F = 91/48 ≈ 1.896. Domain walls form SLE(3) curves with fractal dimension 11/8.
Watch the Binder cumulant sparkline near T_c. It should hover near the universal value U* ≈ 0.6107, indicated by the dashed sanguine guide line. Below T_c it rises toward 2/3; above T_c it falls toward 0. The crossing is how you locate T_c without knowing it in advance — it is the method that lets you measure critical temperatures in materials whose exact solutions you don't have.