Ising model simulation

The Ising model is the simplest model of a magnet: a grid of tiny spins that point up or down and prefer to match their neighbours. It shows how a material suddenly magnetises below a critical temperature, a phase transition.

Interactive, free and runs in your browser. Part of the Thermodynamics simulations.

What you will see

A 40×40 lattice of spins updated with the Metropolis algorithm. At high temperature the spins flicker randomly. Cool it towards the critical temperature and patches of matching spins grow and merge; below it, one direction takes over and the lattice magnetises.

Try this

  1. Start hot and lower the temperature slowly through the critical point.
  2. Watch the domains near the critical temperature: they appear at every size.
  3. Cool it fast and see domains get stuck before one wins.

Common misconception

Below the critical temperature the magnet does not choose a direction because of an outside field. Either direction is equally likely; random fluctuations decide, which is called spontaneous symmetry breaking.

Exam-style question

What is the critical temperature of the 2D square-lattice Ising model?

Onsager’s exact result: kT_c / J = 2 / ln(1 + √2) ≈ 2.269. Above it the lattice is disordered; below it, it magnetises spontaneously.

Key equations

  • E = −J Σ sᵢsⱼ
  • P(flip) = min(1, e^{−ΔE/kT})
  • kT_c / J ≈ 2.269

Curriculum

AQA A-level physics 3.6.2, IB Physics B.1–B.4.

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