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Phase transitions lab · Potts

The Potts model: when more choice makes the change abrupt

Give each site q colours instead of two, and let neighbours care only whether they match. The critical temperature is known exactly for every q. What happens there is not the same for every q: up to four colours, order fades in gradually; past four, it arrives in one jump, the way water freezes.

The model

Each site takes one of q colours. A matching pair of neighbours lowers the energy by one unit; a mismatched pair costs nothing, whichever two colours they are:

E=−∑⟨ij⟩δ(σi,σj)E = -\sum_{\langle ij\rangle} \delta(\sigma_i, \sigma_j)

Renfrey Potts studied it in 1952, following a suggestion from his supervisor Cyril Domb. A duality between low and high temperatures fixes the transition exactly:

Tc=1ln⁡(1+q)T_c = \frac{1}{\ln\left(1 + \sqrt{q}\right)}

With q = 2 this is the Ising model at half the coupling, so Tc ≈ 1.135, half of Ising’s 2.269. The order parameter is the share of the commonest colour, scaled so that it reads 0 when all q colours are equally common and 1 when one colour has taken over.

Continuous or abrupt

Rodney Baxter showed in 1973 that the transition is continuous for q ≤ 4 and first order for q > 4. In a continuous transition the ordered and disordered phases become the same thing at Tc, and the system fluctuates between them at every scale. In a first-order transition they stay different: the energy jumps at Tc, and that jump is a latent heat, like the heat that melts ice without warming it.

At a first-order transition both phases are stable, so either can persist past Tc for a while. That is why water can be supercooled. Choose Eight at Tc: the ordered half and the disordered half sit side by side, and the border between them wanders without either taking over quickly.

Exactly at Tc the average energy per site of the two phases is −(1+1/q)-(1 + 1/\sqrt{q}), known exactly; it is the horizontal line on the energy chart. For q ≤ 4 the curve passes smoothly through it. For q = 8, run a cooling sweep and then a heating sweep with Metropolis moves: the two curves part company around Tc, each lagging in its own phase, with a gap where the latent heat is.

Grain growth

With many colours, cold, the Potts model stops being about a transition and becomes a model of grains: the crystals in a cooling metal, or the bubbles in a foam. Each colour is one crystal orientation. Atoms at a boundary between grains are in a higher-energy state, so the boundaries shrink, and a boundary moves towards the centre of its curvature. A grain with fewer than six neighbours has sides that bulge outwards, so its boundaries close in and it shrinks; a grain with more than six grows.

Choose Grain growth and set the chart to Grain size. The average grain area grows roughly in proportion to time, the law that both metals and foams follow. In two dimensions there is an exact rule for each grain alone, due to John von Neumann (1952): a grain’s area changes at a rate proportional to its number of sides minus six. Six-sided grains hold steady, smaller polygons shrink and vanish.

Two shortcuts make this run: moves only propose a neighbour’s colour (with 48 colours a random one would almost never match), and the temperature is kept a little above zero, because on a square grid at zero temperature many boundaries get stuck. The method itself goes back to Anderson, Srolovitz, Grest and Sahni (1984).

Things to try

  • Compare q = 3 and q = 8, each at its own Tc with Wolff moves. One flickers between order and disorder; the other commits.
  • Quench q = 4 to half its Tc. Where three domains meet, the walls form Y-shaped junctions.
  • In grain growth, paint a big square grain. Its corners round off, and how fast it grows depends on how many neighbours it ends up with.
  • Watch the heat capacity peak sharpen as q rises past 4.

The other models in the lab

New to all this? The lab’s introduction explains order parameters, critical points and how the simulations work.

References