Spins flip by Metropolis sampling — watch spontaneous magnetization emerge below the Curie temperature
Spin lattice (+1 white, -1 black)
Magnetization over time
Energy over time
0.00|Magnetization|
0.00Energy/spin
2.27Temperature T
The 2D Ising model has an exact solution (Onsager 1944): the critical temperature is Tc = 2/ln(1+√2) ≈ 2.269.
Below Tc, spins align into large domains (ferromagnetic order). Above Tc, thermal fluctuations destroy long-range order.
Each Monte Carlo step attempts N random spin flips using the Metropolis criterion: accept if ΔE < 0, else accept with probability e−ΔE/T.