The XY model: a transition with no magnet

Each site holds an arrow at angle $\theta$, drawn as a colour: red points right, and the hue goes once round the colour wheel as the arrow turns. Neighbours prefer to point the same way:

Restrict the angles to q evenly spaced directions and it is the q-state clock model. With two directions the cosine is just ±1 and this is exactly the Ising model. With any angle at all, it is the XY model. The Allowed directions control slides between them.

In the Ising model, turning a region over costs a domain wall, and walls are expensive. With arrows that can turn by any amount, a region can be twisted gently instead: a slow rotation of a few degrees per site costs almost nothing. At any temperature above zero, heat excites such gentle twists at every length scale, and over a large enough distance they add up to a full turn. The average arrow shrinks to zero as the grid grows, however cold it is. This is the Mermin–Wagner theorem (1966): in two dimensions, a continuous symmetry cannot be broken at T > 0.

On a finite grid the average arrow is still long when cold; it just shortens, slowly, as the grid grows. The |M| chart shows it.

Walk round a small square of four sites and add up how much the arrow turns. Usually the total is zero. Sometimes it is a full turn: the colours run once round the wheel as you circle the spot. That is a vortex (white dot) if the arrow turns the same way you walk, and an antivortex (black dot) if it turns the other way.

A vortex cannot be removed by any small adjustment of the arrows around it; it is a topological defect. It can only disappear by meeting an antivortex. After a quench, watch exactly that: pairs drift together and annihilate, and the swirls smooth out.

A lone vortex disturbs the arrows all the way out to the edge of the grid, and its energy grows with the logarithm of the grid size. A vortex–antivortex pair disturbs only the region around it, so its energy grows only with the logarithm of its separation.

A free vortex costs energy that grows like $\ln L$, but it can sit at any of about $L^2$ places, so its entropy grows like $\ln L^2 = 2\ln L$. Both grow the same way, so the balance is decided by the temperature alone. Michael Kosterlitz and David Thouless saw in 1973 that this gives a sharp transition. Below $T_{\mathrm{BKT}} \approx 0.893$, vortices exist only in tightly bound pairs. Above it, the pairs unbind and free vortices scramble the arrows over short distances.

The order parameter is not the magnetisation, which is zero on both sides. It is the spin stiffness $\Upsilon$: how much energy it costs to twist the arrows slowly from one side of the grid to the other. Nelson and Kosterlitz showed in 1977 that on an infinite grid $\Upsilon$ jumps from $2T/\pi$ straight to zero at $T_{\mathrm{BKT}}$. On the chart, the measured stiffness follows its own curve down until it meets the $2T/\pi$ line, then drops. A finite grid smooths the drop, and the crossing sits slightly above the infinite-grid value.

• q = 2 is the Ising model: Tc ≈ 2.269, sharp domain walls.
• q = 3 is the three-state Potts model in disguise, with $T_c = 1.5/\ln(1 + \sqrt{3}) \approx 1.493$.
• q = 4 splits into two independent Ising models, each with half the coupling, so Tc is exactly half of Ising’s, ≈ 1.135.
• q ≥ 5 has two transitions (José, Kadanoff, Kirkpatrick and Nelson, 1977). Below the lower one the arrows lock into q kinds of domain; above the upper one they are disordered; in between, the allowed directions stop mattering and the model behaves like XY, vortices and all. For q = 6 the two transitions are near T ≈ 0.70 and 0.90.

• Choose One pair and watch the vortex and antivortex drift together. Drag on the grid to place more pairs, near and far apart.
• Run a Wolff heating sweep and watch the stiffness chart: the dots fall off the $2T/\pi$ line near 0.9.
• Count vortices with the chart set to Vortices: below about 0.8 there are almost none; above 1 they multiply fast.
• Quench with q = 6 at T = 0.3. Six-coloured domains form, with vortices pinned where all six meet.
• Add a small field h. The arrows pick a direction, and the vortices pair off and vanish much sooner.

The same transition governs thin films of superfluid helium, where the arrow is the phase of the quantum wavefunction; Bishop and Reppy measured the jump in superfluid density in 1978, and it matched the universal value. It also appears in thin superconducting films and in flat clouds of ultracold atoms, where free vortices show up as dislocations in interference images (Hadzibabic and colleagues, 2006).

• Ising model Up-or-down magnets that line up below one exact temperature, and lose it above.
• 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.

• J. M. Kosterlitz and D. J. Thouless (1973), Ordering, metastability and phase transitions in two-dimensional systems, J. Phys. C.
• N. D. Mermin and H. Wagner (1966), Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models, Phys. Rev. Lett..
• D. R. Nelson and J. M. Kosterlitz (1977), Universal Jump in the Superfluid Density of Two-Dimensional Superfluids, Phys. Rev. Lett..
• J. V. José, L. P. Kadanoff, S. Kirkpatrick and D. R. Nelson (1977), Renormalization, vortices, and symmetry-breaking perturbations in the two-dimensional planar model, Phys. Rev. B.
• M. Hasenbusch (2005), The two-dimensional XY model at the transition temperature: a high-precision Monte Carlo study, J. Phys. A; the source of T_BKT ≈ 0.8929.