by Dirk Brockmann

This explorable illustrates pattern formation and dynamics in the $XY$-model, an important model in statistical mechanics for studying phase-transitions and other phenomena. It’s a generalization of the famous Ising-Model. The $XY$-model is actually quite simple.

The model is defined by a two dimensional square lattice. Each lattice site hosts a magnetic dipole, illustrated by the little needles below. They can freely rotate around their central pivot and align with the magnetic field of their surrounding.

Initially all the dipoles are oriented randomly. Press Play and see what happens.

This is how it works

In the model the orientation of a dipole $n$ is quantified by and angle variable $\theta _n \in [ 0, 2\pi ] $. The dipole interacts with its eight nearest neighbors on the lattice and sees and average magnetic field, which is the average of the orientations of the neighbors. Dipole $n$ then slowly aligns its orientation towards the average orientation of its neighbors. Every dipole is doing this at the same time.

Observe this

When you press Play, all the dipoles start rotating until regions of similar orientation emerge in which all dipoles are approximately aligned. However, sometimes you will see points on the lattice in which defects appear where different orientations meet and where they can’t seem to decide which way to orient.

If you are patient and you wait a bit you will see that every now and then, even though the patterns seems frozen, defects will sort themselves out and disappear until everything is almost perfectly aligned.

Continuous Space

These effects become more clearly visible when we go to a larger system that mimics continuous spatial coordinates. Click on continuous and you will see a colorful lattice. Each pixel now is a dipole and the rainbow colorscale encodes the dipoles’ orientation.

If you press play now, you will see that quickly regions of similar hue (orientation) emerge. And, again the defects appear as points around which the orientation varies. You can see that slowly, these defects approach each other and annihilate.

The pattern that emerges is very similar to what you see if you have a system of phase-coupled oscillators on a lattice, discussed in Explorable Spin-Wheels.

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