What does the Metropolis algorithm do here?

At each trial it picks a random lattice site, computes the energy change ΔE of flipping that spin, and accepts the flip unconditionally if ΔE ≤ 0 or with probability e^(−ΔE/kT) if ΔE > 0. Repeating this many times correctly samples the Boltzmann distribution at temperature T.

What happens at the Curie temperature?

Below Tc (≈2.269 J/k for the 2D Ising model) the exchange coupling dominates thermal noise and the lattice spontaneously orders into large aligned domains. Above Tc thermal fluctuations win and the lattice stays disordered with near-zero net magnetisation.

What is the difference between the ferromagnetic and antiferromagnetic settings?

With J = +1 (ferromagnetic) neighbouring spins prefer to align, producing net magnetisation below Tc. With J = −1 (antiferromagnetic) neighbouring spins prefer to anti-align, producing a checkerboard pattern with near-zero net magnetisation even when fully ordered.

Home▸Quantum Physics▸2D Spin Lattice (2D Canvas): Heisenberg Ferromagnet

2D Spin Lattice (2D Canvas): Heisenberg Ferromagnet

Canvas-2D Ising/Heisenberg spin lattice: run Metropolis Monte Carlo below and above the Curie temperature and watch spontaneous magnetisation form and vanish.

Quantum Physics2DModerate60 FPS📱 Mobile-adapted⇄ 3D version

This 2D companion runs the same Ising/Heisenberg spin-lattice Hamiltonian as the 3D version on a plain canvas grid instead of a rendered scene: temperature, exchange coupling J, an external field and the lattice size are all live controls, a Metropolis Monte Carlo loop flips spins with the correct Boltzmann acceptance probability every frame, and a strip chart tracks magnetisation over time so the ferromagnetic phase transition around T ≈ 2.269 J/k is directly visible as the lattice snaps from disordered noise into large aligned domains.

⚙ Under the hood

Canvas-2D Ising/Heisenberg spin lattice with Metropolis Monte Carlo sampling, live magnetisation and energy readouts, and a mouse-paintable seed domain.

ising modelheisenberg modelferromagnetismmetropolis algorithmmonte carlocurie temperature

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install