Magnet Phase Transition LabBeta

About Magnet Phase Transition Lab

Each cell of a square lattice holds a spin that can only point up (+1) or down (−1). Two neighbouring spins that agree lower the energy by 1; two that disagree raise it by 1. An external field h gives an extra advantage of h to spins pointing its way. That is the whole rulebook, yet as you cool the lattice there is a sharp temperature below which the whole grid lines up. It is the barest possible model of iron becoming a magnet below its Curie point.

Temperature T is measured in units where the coupling J and Boltzmann's constant k are both 1. The edges wrap around (periodic boundaries), so the lattice has no edges, like the surface of a doughnut. You can choose 128², 256² or 512² spins; the default is 256².

Onsager's exact critical temperature

In 1944 Lars Onsager solved the zero-field 2D Ising model exactly. On the square lattice the transition happens where sinh(2J/kTc) = 1, which gives Tc = 2/ln(1+√2) ≈ 2.269. That is the tick mark on the temperature slider and the value the go-to-Tc button sets. At this temperature the specific heat diverges logarithmically and the energy per spin is exactly −√2 ≈ −1.414.

Below Tc the spontaneous magnetisation is M = [1 − sinh⁻⁴(2/T)]^(1/8), a result Onsager announced and C. N. Yang proved in 1952. The pair of dashed Onsager lines on the graph sit at +M and −M from this formula. The exponent 1/8 means magnetisation appears very steeply just below Tc. Mean-field theory predicts 1/2, which is wrong in two dimensions.

Experiment 1: domains growing after a quench

Reset to random and set T = 1.5. A random lattice is the infinite-temperature state, so this is a sudden quench. Small patches appear, then small domains are swallowed by large ones and the boundaries shorten. Theory says |M| ≈ 0.987, yet the readout sits near zero for a while because up and down domains of similar size are competing. With periodic boundaries you sometimes get two stripes spanning the lattice that last a very long time. A straight boundary has no curvature pulling it in, so nothing drives it away.

Switch to Wolff clusters in the same situation and big chunks flip in one move, so the lattice orders quickly. Both algorithms sample the same equilibrium; they differ only in how fast they reach it.

Experiment 2: patches of every size at Tc

Press go-to-Tc. At T = 2.269 domains of all sizes are visible at once. The correlation length diverges at the critical point, leaving a fractal-looking pattern with no characteristic scale. Metropolis has to flip large domains one spin at a time, so the picture changes very slowly; this is critical slowing down. Wolff's method bonds aligned neighbours with probability 1 − e^(−2/T) and flips the whole cluster. At Tc that probability is exactly 2 − √2 ≈ 0.586. Wolff's 1989 paper reported that the method almost eliminates critical slowing down for the 2D Ising model.

Grow the lattice from 128 to 512 and |M| at Tc gets a little smaller. Finite-size scaling says the critical magnetisation goes as L^(−1/8), so a lattice four times wider shows about 4^(1/8) ≈ 1.19 times less. On an infinite lattice it would be zero.

Experiment 3: a field, and a state that refuses to flip

Start with all up, set T = 1.5 and h = −0.3. Every spin now opposes the field, so flipping is favourable overall, but under Metropolis the lattice holds out for a long time. Flipping one spin inside the up region breaks four bonds (cost 8) and gains only 0.6 from the field; the acceptance probability is exp(−7.4/1.5) ≈ 0.007. Only when a down seed happens to grow large enough does the down phase spread. It is the same mechanism as supercooled water waiting for a nucleus before it freezes.

Paint a large down disk with the brush and the transition starts at once. Paint a small one and it shrinks away, because its boundary costs more than its interior gains. The size that survives is the critical nucleus.

What the graph is and isn't telling you

The magnetisation plotted is signed. Below Tc a finite lattice can wander between +M and −M, especially with Wolff updates, which is why the Onsager lines are drawn at both +M and −M. Above Tc the theory says zero, but a finite lattice shows small fluctuations of order 1/L in magnitude, larger on the 128² grid than on 512².

Worked examples

How to use it

1. Pick a starting state: random, all up, all down or half and half.
2. Move T and h and watch magnetisation and energy on the graph.
3. Switch the update method between Metropolis and Wolff and compare speed.
4. Paint spins with the brush to test nucleation and domain walls.

Frequently asked questions

Notes

The theory curve is for an infinite lattice. On a finite lattice the transition is rounded near Tc, and |M| fluctuates slightly above zero even above Tc.

There is no exact solution with a nonzero field, so the theory lines are hidden when h ≠ 0.