Three days ago, the 2016 Nobel Prize in Physics was awarded to Michael Kosterlitz of Brown University:
David Thouless of the University of Washington:
and Duncan Haldane of Princeton University:
They won it for their “theoretical discovery of topological phase transitions and topological phases of matter”, which was later confirmed by many experiments.
Sadly, the world’s reaction was aptly summarized by Wired magazine’s headline:
Nobel Prize in Physics Goes to Another Weird Thing Nobody Understands
Journalists worldwide struggled to pronounce ‘topology’, and a member of the Nobel prize committee was reduced to waving around a bagel and a danish to explain what the word means:
That’s fine as far as it goes: I’m all for using food items to explain advanced math! However, it doesn’t explain what Kosterlitz, Thouless and Haldane actually did. I think a 3-minute video with the right animations would make the beauty of their work perfectly clear. I can see it in my head. Alas, I don’t have the skill to make those animations—hence this short article.
I’ll just explain the Kosterlitz–Thouless transition, which is an effect that shows up in thin films of magnetic material. Haldane’s work on magnetic wires is related, but it deserves a separate story.
I’m going to keep this very quick! For more details, try this excellent blog article:
• Brian Skinner, Samuel Beckett’s guide to particles and antiparticles, Ribbonfarm, 24 September 2015.
I’m taking all my pictures from there.
Imagine a thin film of stuff where each atom’s spin likes to point in the same direction as its neighbors. Also suppose that each spin must point in the plane of the material.
Your stuff will be happiest when all its spins are lined up, like this:
What does ‘happy’ mean? Physicists often talk this way. It sounds odd, but it means something precise: it means that the energy is low. When your stuff is very cold, its energy will be as low as possible, so the spins will line up.
When you heat up your thin film, it gets a bit more energy, so the spins can do more interesting things.
Here’s one interesting possibility, called a ‘vortex’:
The spins swirl around like the flow of water in a whirlpool. Each spin is fairly close to being lined up to its neighbors, except near the middle where they’re doing a terrible job.
The total energy of a vortex is enormous. The reason is not the problem at the middle, which certainly contributes some energy. The reason is that ‘fairly’ close is not good enough. The spins fail to perfectly line up with their neighbors even far away from the middle of this picture. This problem is bad enough to make the energy huge. (In fact, the energy would be infinite if our thin film of material went on forever.)
So, even if you heat up your substance, there won’t be enough energy to make many vortices. This made people think vortices were irrelevant.
But there’s another possibility, called an ‘antivortex’:
A single antivortex has a huge energy, just like a vortex. So again, it might seem antivortices are irrelevant if you’re wondering what your stuff will do when it has just a little energy.
But here’s what Kosterlitz and Thouless noticed: the combination of a vortex together with an antivortex has much less energy than either one alone! So, when your thin film of stuff is hot enough, the spins will form ‘vortex-antivortex pairs’.
Brian Skinner has made a beautiful animation showing how this happens. A vortex-antivortex pair can appear out of nothing:
… and then disappear again!
Thanks to this process, at low temperatures our thin film will contain a dilute ‘gas’ of vortex-antivortex pairs. Each vortex will stick to an antivortex, since it takes a lot of energy to separate them. These vortex-antivortex pairs act a bit like particles: they move around, bump into each other, and so on. But unlike most ordinary particles, they can appear out of nothing, or disappear, in the process shown above!
As you heat up the thin film, you get more and more vortex-antivortex pairs, since there’s more energy available to create them. But here’s the really surprising thing. Kosterlitz and Thouless showed that as you turn up the heat, there’s a certain temperature at which the vortex-antivortex pairs suddenly ‘unbind’ and break apart!
Why? Because at this point, the density of vortex-antivortex pairs is so high, and they’re bumping into each other so much, that we can’t tell which vortex is the partner of which antivortex. All we’ve got is a thick soup of vortices and antivortices!
What’s interesting is that this happens suddenly at some particular temperature. It’s a bit like how ice suddenly turns into liquid water when it warms above its melting point. A sudden change in behavior like this is called a phase transition.
So, the Kosterlitz–Thouless transition is the sudden unbinding of the vortex-antivortex pairs as you heat up a thin film of stuff where the spins are confined to a plane and they like to line up.
In fact, the pictures above are relevant to many other situations, like thin films of superconductive materials. So, these too can exhibit a Kosterlitz–Thouless transition. Indeed, the work of Kosterlitz and Thouless was the key that unlocked a treasure room full of strange new states of matter, called ‘topological phases’. But this is another story.
What is the actual definition of a vortex or antivortex? As you march around either one and look at the little arrows, the arrows turn around—one full turn. It’s a vortex if when you walk around it clockwise the little arrows make a full turn clockwise:
It’s an antivortex if when you walk around it clockwise the little arrows make a full turn counterclockwise:
Topologists would say the vortex has ‘winding number’ 1, while the antivortex has winding number -1.
In the physics, the winding number is very important. Any collection of vortex-antivortex pairs has winding number 0, and Kosterlitz and Thouless showed that situations with winding number 0 are the only ones with small enough energy to be important for a large thin film at rather low temperatures.
Now for the puzzles:
Puzzle 1: What’s the mirror image of a vortex? A vortex, or an antivortex?
Puzzle 2: What’s the mirror image of an antivortex?
Here are some clues, drawn by the science fiction writer Greg Egan:
and the mathematician Simon Willerton:
To dig a bit deeper, try this:
• The Nobel Prize in Physics 2016, Topological phase transitions and topological phases of matter.
It’s a very well-written summary of what Kosterlitz, Thouless and Haldane did.
Also, check out Simon Burton‘s simulation of the system Kosterlitz and Thouless were studying:
In this simulation the spins start out at random and then evolve towards equilibrium at a temperature far below the Kosterlitz–Thouless transition. When equilibrium is reached, we have a gas of vortex-antivortex pairs. Vortices are labeled in blue while antivortices are green (though this is not totally accurate because the lattice is discrete). Burton says that if we raise the temperature to the Kosterlitz–Thouless transition, the movie becomes ‘a big mess’. That’s just what we’d expect as the vortex-antivortex pairs unbind.
I thank Greg Egan, Simon Burton, Brian Skinner, Simon Willerton and Haitao Zhang, whose work made this blog article infinitely better than it otherwise would be.