🧲 2D Ising Model

Tc ≈ 2.269 J/kB (Onsager, exact 2D solution)
Temperature T2.270
Magnetisation m0.000
Energy / spin0.000
Paramagnet
Click or drag on the lattice to flip spins by hand. E = −J Σ sisj − h Σ si, summed over nearest neighbours. Metropolis acceptance: min(1, exp(−ΔE/T)).

🧲 2D Ising Model

The 2D Ising model is the canonical lattice model of ferromagnetism: a square grid of spins, each ±1, interacting only with their four nearest neighbours. It is the textbook example of a phase transition and one of the very few interacting models solved exactly (Onsager, 1944), giving the critical temperature Tc = 2J / (kB ln(1+√2)) ≈ 2.269 J/kB.

Each Metropolis step picks a random spin, computes ΔE from its four neighbours and the field h, and flips it if ΔE ≤ 0, or with probability exp(−ΔE/T) otherwise. Many such attempts per frame drive the lattice toward thermal equilibrium at the chosen temperature.

Below Tc the spins spontaneously align into large pink/blue domains — a ferromagnet. Above Tc thermal noise wins and the lattice looks like static — a paramagnet.