Ising Model
A 200 × 200 lattice of binary magnetic spins evolved by Metropolis–Hastings Monte Carlo. Ferromagnetic domains compete and coarsen below the critical temperature; thermal disorder destroys long-range order above it. At the Onsager critical point — Tc ≈ 2.269 J/kB — the domain pattern is scale-free, fractal, and governed by universal exponents that appear in systems ranging from magnets to protein folding to the early universe. Onsager's exact solution (1944) was one of the landmark results of 20th-century theoretical physics.
The Hamiltonian
Each lattice site i carries a spin sᵢ ∈ {−1, +1}. Nearest-neighbour pairs interact ferromagnetically, and an optional external field h breaks the spin-flip symmetry:
H = −J Σ⟨ij⟩ sᵢ sⱼ − h Σᵢ sᵢ
With J > 0 (ferromagnetic), aligned neighbours lower the energy. Entropy opposes this at finite temperature — the competition between energy and entropy is the engine of the phase transition. Periodic boundary conditions eliminate surface effects and let the lattice approximate the thermodynamic limit.
Metropolis–Hastings algorithm
At each Monte Carlo step a site is chosen at random and a spin flip is proposed. The energy cost of the flip is ΔE = 2sᵢ (J Σj∈nn sⱼ + h), where the sum runs over the four nearest neighbours. The flip is accepted with probability
P(accept) = min(1, e−ΔE / kBT)
This satisfies detailed balance, guaranteeing that the long-run distribution converges to the Boltzmann measure at temperature T. Energetically favourable flips (ΔE ≤ 0) are always accepted; unfavourable ones are accepted stochastically, allowing the system to escape local minima. One sweep = N² attempted flips.
Onsager solution & the critical point
Lars Onsager's 1944 exact solution of the 2D square-lattice Ising model remains one of the great achievements of theoretical physics. The free energy is analytic everywhere except at the critical temperature, where it has a logarithmic singularity:
Tc = 2J / (kB ln(1 + √2)) ≈ 2.2692 J/kB
Below Tc, spontaneous magnetisation persists even at h = 0 — the Z₂ spin-flip symmetry is spontaneously broken and the system selects one of two ground states. Above Tc, only paramagnetic disorder survives. The transition is continuous (second-order): the magnetisation vanishes as |M| ∼ (Tc − T)β with β = 1/8, and the correlation length diverges as ξ ∼ |T − Tc|−1. At Tc itself, spin clusters appear at every length scale — the lattice is statistically self-similar.
What to look for
Hit Ordered to start from a perfectly aligned cold state, then drag the temperature up through Tc ≈ 2.27: watch domains fragment until they lose all coherence above 3. Hit Critical to land directly on the transition — the blue and red regions then form a fractal mosaic with no preferred scale. Apply a positive field to watch the red (spin-down) minority phase shrink and vanish, then flip the field sign to reverse the process. The AFM initial state places the system in a checker-board antiferromagnetic configuration — at low T this is a metastable trap the system slowly escapes through thermal activation.