The Ising model: a magnet in one rule

Each site i holds a spin $s_i = +1$ (up, gold) or $-1$ (down, dark). Each pair of neighbours that agree lowers the energy by one unit; each pair that disagree raises it by one. An outside field h adds a preference for one direction:

At temperature T, a configuration turns up with probability proportional to $e^{-E/T}$. Cold, the lowest energy wins: everything aligned. Hot, there are vastly more disordered configurations than ordered ones, and their sheer number wins. The magnetisation M, the average spin, measures which side is winning.

Ernst Ising solved the one-dimensional chain in 1925 and found no transition at all, and guessed that two dimensions would be no different. He was wrong. In 1944 Lars Onsager solved the two-dimensional model exactly with no field, and found a sharp transition at

Below Tc the magnetisation is $M = \left(1 - \sinh^{-4}(2/T)\right)^{1/8}$: Onsager announced this in 1948 and C. N. Yang proved it in 1952. It is the dashed curve on the chart. Near Tc it falls like $(T_c - T)^{1/8}$, an exponent no simple theory had predicted. The exact energy and heat capacity are also on the chart; the heat capacity has a spike at Tc that grows with the logarithm of the grid size.

• Below Tc, after a quench, domains of each colour form at once and then grow. A domain wall costs energy for every unlike pair along it, so walls shorten and straighten, and small domains shrink away. The typical domain size grows like the square root of time.
• At Tc, domains come in every size, with islands inside islands inside islands. Zoom your eyes out and the picture looks the same. This is the hallmark of a continuous transition, and it is where the 1/8 comes from.
• Above Tc, small patches still agree, because neighbours still lean together, but the patches have a typical size that shrinks as T rises.

Press Cool to step the temperature from 4 down to 1, letting the grid settle at each value and then averaging |M|. With Wolff moves the dots land on Onsager’s curve below Tc, except within a whisker of Tc, where a finite grid always rounds the corner. Above Tc they sit a little above zero, because a finite grid is never perfectly balanced.

With Metropolis moves, cooling can go wrong: the grid can freeze into two broad stripes, one up and one down, with straight walls that cost almost nothing to keep. |M| stays near zero although every region is ordered. Real magnets do this too; it is one reason a lump of iron is not a magnet until you magnetise it.

The χ and heat capacity charts come from the same sweep: they are the size of the fluctuations in M and E. Both peak at Tc, where the grid cannot make up its mind.

Press Clusters. Neighbours with the same spin are joined at random with probability $1 - e^{-2/T}$, and the connected pieces are coloured. These are the Fortuin–Kasteleyn clusters, and they are exactly what a Wolff move flips. Below Tc one of them spans the grid (drawn white); above Tc none does; and they first span at precisely Tc. The magnetic transition is a percolation transition of these clusters, which is why the forest fire has the same shape of threshold.

Plain same-coloured patches also first span at Tc in two dimensions, but their sizes follow the wrong power laws. The randomly bonded clusters are the ones whose statistics match the magnet’s.

Below Tc, drag the field h against the magnetisation, or press the hysteresis preset. The magnet does not flip when h crosses zero. A domain of the other sign has to form first, and a small one costs more in wall than it gains from the field, so the flip waits until the field is well past zero. On the way back the same thing happens in reverse, and M against h traces a loop. The loop’s width depends on how fast you sweep, because forming a big enough domain takes time.

With Kawasaki moves, sites swap with a neighbour instead of flipping, so the number of up spins never changes. Read up as one liquid and down as another that would rather not mix: cooled, they separate into blobs. Material has to travel through the other phase to move, so the blobs grow more slowly than magnetic domains, like $t^{1/3}$ instead of $t^{1/2}$. With a quarter of the sites up, the minority forms round droplets, and large droplets grow while small ones evaporate.

• Run the same cooling sweep on the small and the large grid. Where do they differ most?
• At Tc, switch from Wolff to Metropolis and watch how much slower the picture changes. That is critical slowing down.
• Below Tc, paint a large down circle into an up magnet. It shrinks and disappears, at a rate set by its curvature.
• Set h to a small positive value above Tc. The transition disappears into a smooth crossover.

Real magnets have spins that can point in any direction, interactions beyond nearest neighbours, and three dimensions. The two-dimensional Ising model is still accurate near Tc for magnets made of weakly coupled layers with a strong preferred axis, such as K₂CoF₄. More widely, it is the lattice gas: a model of a fluid in which each site is filled or empty, and the critical point of a liquid and its vapour belongs to the Ising universality class.

• 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.

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

• L. Onsager (1944), Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition, Phys. Rev..
• C. N. Yang (1952), The Spontaneous Magnetization of a Two-Dimensional Ising Model, Phys. Rev..
• U. Wolff (1989), Collective Monte Carlo Updating for Spin Systems, Phys. Rev. Lett..
• K. Kawasaki (1966), Diffusion Constants near the Critical Point for Time-Dependent Ising Models. I, Phys. Rev..
• B. M. McCoy and T. T. Wu, The Two-Dimensional Ising Model, Harvard University Press (1973); the exact results in full.