Phase transitions: where order switches on

The lab opens on the Ising magnet, the simplest and best understood. The others change one thing about it at a time, and each change teaches something new.

• Ising model Up-or-down magnets that line up below one exact temperature, and lose it above.
• Clock and XY models Arrows that can point any way. No magnet forms, yet vortices pair up below T_BKT.
• Potts model q colours instead of two. Past four, the transition stops being gradual.
• Percolation and forest fires Plant trees at random and light one edge. Past 59.27% the fire gets across.
• Avalanches and crackling noise Disordered magnets flip in bursts of every size as the field rises.
• Hopfield memory An Ising model whose couplings store pictures, and roll a damaged one back.

Each model has a quantity that is zero in the disordered phase and not zero in the ordered one, called its order parameter. For a magnet it is the magnetisation: the average of all the little magnets, which cancels out when they point every which way. For a forest it is the share of trees in the one cluster that reaches across.

The order parameter is what the charts plot. Far from the transition it changes slowly. Near the critical point it changes fast, and on an infinite grid its slope becomes infinite. A real grid has a finite number of sites, which rounds the corner off. Change the grid size and watch the rounding change.

Right at a continuous transition, the regions that happen to agree come in every size, from a single site up to the whole grid, and the picture has no typical length. Zoom out and it looks statistically the same. That is why the Ising magnet at Tc, the forest at p_c and the three-colour Potts model at its Tc all look like islands within islands.

It also makes the details stop mattering. Near the critical point, very different systems behave alike if they share a dimension and a symmetry. A carbon dioxide liquid–gas critical point and a uniaxial magnet, both three-dimensional, follow the same power laws with the same exponents. This is called universality, and it is why a model as crude as the Ising magnet makes accurate predictions about real materials.

Most of the models are thermal: a configuration with energy E turns up with probability proportional to $e^{-E/T}$. Listing every configuration is impossible: a 128 × 128 grid of up-or-down spins has $2^{16384}$ of them. Instead the lab wanders between configurations with Monte Carlo moves chosen so that, in the long run, each one is visited with the right probability.

The Metropolis move (Metropolis and colleagues, 1953) picks a site and proposes a change. If the change lowers the energy, it happens. If it raises the energy by $\Delta E$, it happens with probability $e^{-\Delta E/T}$. One sweep is one attempt per site on average. The Wolff move (1989) builds a cluster of aligned sites and flips it in one go. Near a critical point, regions are huge and Metropolis needs a very long time to turn one over. That slowdown is called critical slowing down, and cluster moves avoid it.

Percolation has no energy at all: trees are planted at random. Avalanches and Hopfield memory run at or near zero temperature, where each spin simply follows the field it feels.

• J. P. Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity, Oxford University Press, with chapters on Monte Carlo and on continuous transitions.
• N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller (1953), Equation of State Calculations by Fast Computing Machines, J. Chem. Phys..
• U. Wolff (1989), Collective Monte Carlo Updating for Spin Systems, Phys. Rev. Lett..
• M. E. J. Newman and G. T. Barkema, Monte Carlo Methods in Statistical Physics, Oxford University Press (1999).