## Kosterlitz–Thouless Transition

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.

### The idea

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.

### Puzzles

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:

### For more

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.

### 79 Responses to Kosterlitz–Thouless Transition

1. stasheff says:

Thanks.

• stasheff says:

That’s fine as far as it goes, but how about enlightening us about the relevance of braid group reps?

• John Baez says:

That’s another story! For what Kosterlitz, Haldane and Thouless did, this is very good:

• The Nobel Prize in Physics 2016, Topological phase transitions and topological phases of matter.

You won’t see any braid group representations in here. However, you’ll see how they laid the groundwork, by showing how the first Chern class and U(1) Chern–Simons theory are important in 1-dimensional and 2-dimensional magnetic materials.

2. Thank you!

3. kram1032 says:

What do you mean by “mirror image”?
As far as I can tell from these pictures, mirroring a vortex will just cause it to change direction, i.e. it will still be a vortex but, if you look at it from above the thin film and it spun around clockwise, after mirroring (or, which should be equivalent, changing your perspective to looking at the same film from below), you’ll get a vortex which spins counterclockwise and vice versa.

Now for the anti-vortex, things seem a little bit more complicated. I can seemingly identify four different axes – a) two along the diagonals, b) two as far off them as possible – with two different outcomes:
Mirroring along a) will not change the system at all.
Mirroring along b) will also reverse direction.
And I’m currently not able to imagine what happens if you look at the same structure from below but I think it might end up reversing the direction just like for normal vortices.

Here is another puzzle: What about higher winding numbers?
I tried scribbling on paper and I think I found a shape that basically traces out a lemniscate which should correspond to a vortex of winding number 2.
I think I, by a similar process, also found something with winding number -2 which seems to look like the above anti-vortex in its hyperbolic-plane-y pattern but with a total of six spokes instead of four, if that makes sense.

Do these also occur? Can it happen, that you, for instance, spontaneously get two anti-vortices and a 2-vortex, where the 2-vortex stays in place and the two anti-vortices fly apart?

Or alternatively, once you are in this new phase where these things become common, will two vortices or anti-vortices merge to become 2-vortices or -2-vortices respectively?

• John Baez says:

kram1032 wrote:

What do you mean by “mirror image”?

I mean that we take the picture, hold it in front of us facing away, and look at its image in a mirror. For example, the mirror image of this:

is this:

• So then it’s exactly what I worked out in my reply above.

• John Baez says:

Oh, so “kram1032” is you! Hi!

Yes, your solutions seem correct. I just didn’t want to give away the answers prematurely.

One can certainly draw vector fields with larger winding number, either positive or negative. I’m fairly sure that, like the basic vortex and antivortex configurations, these ‘n-vortices’ will have infinite energy unless n is zero.

I don’t really know anything about these n-vortices. I would guess that below the Kosterlitz-Thouless transition there’s a pretty good description of the system as a gas of vortex-antivortex pairs, and above it as a dense plasma of vortices and antivortices, without any need to consider n-vortices with |n| > 1. However, I could easily be wrong! I haven’t read any of the extensive literature on this subject!

• Oh whoops I forgot my name differs in g+, heh. Didn’t even think about how that could not have been clear, sorry.

4. Linda A. says:

Thank you for the excellent explanation!

5. anitachowdry says:

Fascinating images.

• John Baez says:

Thanks. But we should thank Richard Skinner!

There are many more images that should be included here, which I can see in my mind. Let me describe two:

1) Start with this:

Then, rotate each arrow about its tail at the same rate. At any moment in time we’ll have a picture that physicists would call a vortex—but they’ll look dramatically different.

2) Start with this:

Then, rotate each arrow about its tail at the same rate. At any moment in time we’ll have a picture that physicists would call an antivortex—but they’ll look dramatically different.

In case anyone wants to animate these, here’s the math. In the first case, at each point with polar coordinates $(r,\theta) = (r_0, \theta_0)$ we draw a unit vector pointing in the direction

$\theta = \theta_0 + t$

at time $t.$ In the second case, at each point with polar coordinates $(r,\theta) = (r_0, \theta_0)$ we draw a unit vector pointing in the direction

$\theta = -\theta_0 + t$

at time $t.$

• Simon Burton says:

Ah, so this is important because we are supposed to think of these vortices as being somehow the same: it’s the U(1) gauge degree of freedom.

• John Baez says:

Right, in the Landau–Ginzburg model of a 2d superconductor, the field usually called

$\psi \colon \mathbb{R}^2 \to \mathbb{C}$

is a unit complex number at the minimum of its potential, so we get a field of unit vectors on the plane. Rotating all these vectors by the same angle counts as a ‘global gauge transformation’: that is, we get a new field that counts as physically equivalent.

We can also take a continuum limit of the classical XY model, and get a simple model of a 2d ferromagnet in which the atom’s spins are described by a field of unit vectors on the plane, which we can think of as a field of unit complex numbers:

$\psi \colon \mathbb{R}^2 \to \mathbb{C}$

and the energy is

$\displaystyle{ \int \int |\nabla \psi|^2 \; dx dy }$

Rotating all the these unit complex numbers by the same angle gives a new field with the same energy. So, it’s a symmetry of the theory!

This symmetry also preserves the ‘winding number’, which is computed as an integral around a large circle in the plane: it’s a way to precisely define the number of vortices minus the number of antivortices.

• Simon Burton says:

The negative vortices look like the gradient of a scalar field with a saddle point, and using the U(1) symmetry we can transform the positive vortices into the gradient of a local minima or maxima. What’s going on here? If I keep thinking about this does it lead to something fancy like Chern-Simons theory?

• John Baez says:

Chern–Simons theory is certainly related to the work of Kosterlitz, Thouless and especially Haldane. Only U(1) Chern–Simons theory shows up, and that makes it a lot simpler than if we had a nonabelian gauge group. The easiest way to get started is to read this:

• The Nobel Prize in Physics 2016, Topological phase transitions and topological phases of matter.

It’s really quite well-written.

The connection to saddle points, local maxima and local minima is simpler. For that, try this:

• Wikipedia, Index of a vector field.

• Greg Egan says:

Here’s an animation of the rotating version John described:

• this looks really neat!
It’s a little weird, though, that the arrows seem to not only rotate, they also shift around.

• John Baez says:

I think that “shifting” is an optical illusion. Knowing Greg, he would have firmly nailed the tail of each arrow to a fixed location. The tails would still seem to shift, since our eyes’ main way of locating an object is relative to other objects, and the heads of the arrows are moving around.

• Is there a specific reaon for rotating the arrows around their tail? Greg’s animations made me dizzy! Here’s my attempt where I rotate the arrows around their centre.

• John Baez says:

Nice! Okay, now I get it. Rotating the arrows around their midpoints rather than their tails is less creepy—it feels less like a drug trip scene in an old 1950’s movie, where the walls are bending.

• Here’s the anti-vortex version with arrows spinning around their centre.

• John Baez says:

Thanks, Simon. Very nice!

Puzzle 3: As these unit vectors rotate, at what moments do we get a vector field that’s obtained by normalizing a vector field with curl zero?

Puzzle 4: As these unit vectors rotate, at what moments do we get a vector field that’s obtained by normalizing a vector field with curl zero?

I thought of these puzzles because at certain moments, these unit vector fields are obtained by normalizing rather famous curl-free vector fields. I don’t know of any deep significance, but there could be one.

• Pramodh Senarath Yapa says:

Hey Simon, can I ask how you made this animation? I want to make a few gifs like this, but I don’t know which program would be best to use. Is there anything you could recommend? Thanks!

• I used maple because that’s what I’ve been using for many years and find it second nature (although I’ve been trying to switch to SageMath of late). You can find my maple code here:
http://simonwillerton.staff.shef.ac.uk/ncafe/vortices.mpl
It ought to be reasonably (!) self-explanatory, but do ask if you want anything explaining.

6. Very interesting.

Can you say some words about what your cobordism hypothesis is (or was before it became a theorem) and whether and how it fits in here?

• John Baez says:

The cobordism hypothesis arose from attempts to classify topological quantum field theories. Very roughly, a field theory is called “topological” if it’s invariant under all diffeomorphisms of spacetime. The theory studied by Kosterlitz and Thouless is not topological in this sense. However, it introduced topological considerations into condensed matter physics in a way that ultimately led to a very fruitful interaction between that subject and topological quantum field theory.

As for what the cobordism hypothesis is, my best explanation is here:

• John Baez and James Dolan, Higher-dimensional algebra and topological quantum field theory, Jour. Math. Phys. 36 (1995), 6073–6105.

• Ah …. so many meanings of “topological”. Thank you for the clarification.

• John Baez says:

It turns out that the most exciting qualitative properties of topological phases are often summarized by topological quantum field theories, even for phases arising from theories that are not themselves topological.

7. theo says:

Looking at the mirror for helical vortices/antivortices is the same as rotating the picture around x or y axes. So here, parity is changed by a 180 deg rotation.

8. Simon Burton says:

I had a bit of fun trying to simulate this system. Here is what I ended up with:

Positive vortices are blue, and negative vortices are green (it’s not totally accurate because the lattice is discrete.) It looks like vortices attract, which makes sense given your description of the infinite energy of an isolated vortex. The temperature is way below the Kosterlitz–Thouless transition. Raising the temperature to that point it just becomes a big mess.

• John Baez says:

Excellent! Yes, a vortex will attract an antivortex, since their total energy goes to infinity as you pull them apart taking their distance to infinity.

Now that I think of it, this is similar to how a quark-antiquark pair in a meson increases in energy roughly linearly as you pull them apart. In that situation the field energy is concentrated in a ‘flux tube’ connecting the quarks. Eventually there’s enough energy to create a new quark-antiquark pair: the flux tube gets cut into two shorter flux tubes.

(I don’t have a good mental image of where the field energy lies when you have a distant vortex-antivortex pair, so I don’t know if it mainly lies in a ‘tube’ connecting the two.)

I wonder if there’s visible dramatic change near the Kosterlitz–Thouless transition. Google leads me to this picture:

but I don’t know exactly what it means. The lower right looks like the ‘big mess’ you mentioned.

• Simon Burton says:

I did some more experimenting and came up with this:

I added a “heat map” to show how much energy each compass needle has, and also varied the temperature so that you can see the onset of the KT transition. It does look like there are tendrils of excitation extending between the vortices, but it’s hard to see.

• John Baez says:

Nice! One can start to see how fascinating this model is. It’s on that nice cusp between being simple enough to understand and complex enough to be deeply interesting.

• Nice! Can you try if a temperature gradient across the critical temperature makes a difference?
Also, do you think you could host a video file somewhere? YouTube makes quite a mess out of the video due to compression. It really doesn’t like lots of small, fast-changing objects.

• Great animation. Somehow I find this vortex-antivortex eating up business similar to the actions in game of life. https://bitstorm.org/gameoflife/

9. Bill says:

Many thanks for your fascinating insights on this phenomenon, John. Please forgive my naivete, but what minimum temperature is required to spontaneously create a vortex-antivortex pair? Can it occur at 0 K? Is there any connection to the creation of electron-positron pairs?

• John Baez says:

At absolute zero any physical system will definitely be in its lowest-energy state, or a randomly chosen lowest-energy state if there is more than one. Since a vortex-antivortex pair has more energy than a state in which all the spins are aligned, you’ll never see a vortex-antivortex pair at absolute zero.

For any physical system as soon as the temperature exceeds absolute zero, there’s some nonzero probability to find that system in any state. Thus, there’s some nonzero probability to find a vortex-antivortex pair in any given large region of the magnet as soon as the temperature exceeds zero, but this probability will be low for low temperatures.

Kosterlitz and Thouless won the Nobel for discovering that this particular system has a phase transition at some positive temperature $T_0.$ This means that the probability of finding vortex-antivortex pair in any given large region of the magnet is not analytic as a function of temperature at the point $T = T_0.$ The system ‘changes phase’, a bit like ice melting or water vaporizing: vortex-antivortex pairs become common and the system is qualitatively different.

This is the famed Kosterlitz–Thouless transition.

Is there any connection to the creation of electron-positron pairs?

It’s slightly similar: as you heat up any system, high-energy states become more probable. Thus as you heat up a box of gas, or a box of radiation, to a sufficiently high temperature, electron-positron pairs will eventually become common. I happen to have worked out the temperature a few weeks ago, and it’s roughly 12 billion kelvin.

To estimate this, take the mass of the electron and convert it into units of energy. The answer is 0.511 MeV, where MeV or ‘million electron volts’ is a common unit of energy in particle physics. Multiply by two to get the energy of a positron-electron pair: roughly 1 MeV. Convert this to a temperature using the formula $E = k_BT$ where $k_B$ is Boltzmann’s constant. In other words, let $T = E/k_B.$ It’s convenient to note that $1/k_B$ is about 12,000 kelvin per eV. Thus, 1 MeV divided by Boltzmann’s constant is about 12,000,000,000 kelvin.

So, at a temperature of about 12 billion kelvin, electron-positron pairs become common. This is not an all-or-nothing thing, and I don’t know if there’s a phase transition. Someone must know.

Temperatures were this high in the early Universe, and also in the jets produced by black holes, and some other extreme situations. Here’s a jet coming out of the supermassive black hole at the center of the galaxy M87, where this is happening:

If I knew the energy of a vortex-antivortex pair, I could do a similar calculation to give a rough estimate of the temperature at which the Kosterlitz–Thouless transition occurs. But I’m sure people have worked out the details in a much more precise: there’s even a whole book on the Kosterlitz-Thouless transition.

10. @whut says:

These curl equations are fascinating and are of course endemic in applications from electromagnetics to fluid dynamics. Perhaps there is some overlap with the model of the QBO equatorial winds that we are working on at John’s Azimuth Forum (see sidebar) and at my blog. Have some notes here:

http://contextearth.com/2016/09/23/compact-qbo-derviation/#comment-199906

Paul Pukite

• @whut says:

A closely related topological phase transition is described by the Quantum Hall Effect and Fractional Quantum Hall Effect. That was a hot topic when I was in grad school growing GaAs quantum well structures. Stormer and Tsui eventually won a Nobel Prize after they experimentally discovered the fractional effect that Laughlin had predicted.

I mention this because in last month’s blog post linked in my comment above, I stated that I applied knowledge of the Hall effect in solving the QBO problem. The two phenomenon share similar curl terms. The math is the common tie between these systems.

“– The perturbation theory is useful when there is a small dimensionless parameter in the problem, and the system is exactly solvable when the small parameter is sent to zero.
– … it is not required that the system has a small parameter,
nor that the system is exactly solvable in a certain limit. Therefore it has been useful in studying strongly correlated
systems, such as the fractional Quantum Hall effect.”

Making progress in one area certainly has application in other disciplines.

11. Fascinating. So how about three dimensions? I guess point like singularities are out of the question if the analogy with vortices still holds. Is there such a thing as an anti vortex ring?

• John Baez says:

In a Type II superconductor, the magnetic field is confined in ‘flux tubes’, also known as Abrikosov vortices. They act like vortice in the following sense: the vector potential $A$ is a vector field with

$\nabla \times A = B$

where $B$ is the magnetic field, but $B = 0$ except inside these tubes. Thus, the $A$ is curl-free outside these tubes— but it curls around inside these tubes! So, you can roughly imagine a flux tube as a little tornado but where the wind is the vector potential $A.$

But it gets better. Thanks to quantum mechanics, each tube carries one ‘quantum of magnetic flux’. That is, the integral of the $B$ field over a disk cutting one of these tubes is a certain fixed value depending only on fundamental constants of nature: the magnetic flux quantum!

If you have a fairly thin film made of a type II superconductor, these flux tubes tend to line up at right angles to the film. In a homogeneous material, they’re happiest forming a lattice! Here’s an actual picture of what they look like, created in a lab in 1989:

So far this is all sounding fairly similar to what Kosterlitz and Thouless were studying. However, I believe that in this situation, the antiparticle of a vortex is a vortex spinning the other way—that is, a mirror-image vortex.

12. Haitao Zhang says:

John,

Your wording is a bit confusing about the appearance of the KT pairs. As Simon Burton’s excellent video shows they appear at far below the transition temperature $T_{\mathrm{KT}}$, as the pairing reduces the activation energy to a finite number. However they stay bound together below $T_{\mathrm{KT}}$ and only break at above $T_{\mathrm {KT}}$ (or in Simon’s words “become a big mess”). This is what the Nobel committee has to say:

The insight by Kosterlitz and Thouless was that at a certain temperature $T_{\mathrm{KT}}$ the pairs will break up into individual vortices. It is this vortex pair unbinding transition that will take the system to a high temperature phase with exponentially decaying correlations.”

However in your blog and comments you termed it as “the Kosterlitz–Thouless transition is the sudden appearance of large numbers of vortex-antivortex pairs as you heat up”, which while maybe true (in the sense that more pairs certainly will appear as the temperature rises) is not quite the same as what the Nobel committee is saying. So which one is correct?

Thanks for the excellent illustrations (and amazing contributions from commenters).

• John Baez says:

The Nobel committee is, unsurprisingly, correct. In my blog article I was trying to say, in a very nontechnical way, that various quantities depend on the temperature in a way that fails to be analytic (i.e., given by a convergent Taylor series) at the phase transition. I guessed that the density of vortices is one of these quantities. I do not actually know this to be true. Since I was in a rush, I made the mistake of stating my guess as a fact. Then I made the further mistake of trying to express my guess in simple terms by saying there’s a “sudden appearance of large numbers of vortex-antivortex pairs” as the temperature passes through the phase transition.

I knew full well that the vortex density could fail to be non-analytic even without a sudden increase in the density as we increase the temperature past the phase transition. I felt mildly guilty about this, but you’ve simultaneously pointed out the problem and pointed to a good expository solution—that is, a nice way to explain what’s really happening!

Briefly: in the Kosterlitz–Thouless transition, the vortex-antivortex pairs “unbind” and we get a messy soup of vortices and antivortices.

This reminds me a bit of a quark-gluon plasma. In neutronium that’s not too hot, quarks, antiquarks and gluons are bound into nucleons, perhaps with some mesons hopping around between them. But when we heat it up enough, we go through a phase transition called ‘deconfinement’, leading to a new state of matter called a quark-gluon plasma. In fact the full story here is more complicated, more interesting, and not fully understood. Click on the image for more:

So, I’ll fix my blog article. However, I would also love to see a graph of the density of

• Haitao Zhang says:

Great. Thanks for the quark-gluon plasma analogy. Very interesting! BTW, I remember reading your physics newsgroup posts before there was the web in the early 90s when I was a physics grad student. I couldn’t really follow back then but I am glad that I can understand this post and was even able to contribute a little.

P.S. Thanks for fixing the formatting issues in my comment. And please feel free to delete my other duplicate comment (which I wasn’t sure was going through).

• John Baez says:

I’m glad you survived my physics posts from the early 90’s! It’s nice to hear from people who remember that ancient era. Thanks again for helping me improve my article. It’s a lot better now, and I think it’s still comprehensible by ordinary folks.

13. Thank you for a very nice explanation of the KT transition. Most of the time I don’t have a clue about who the Nobel prize winners are, but this is something that I actually knew something about, even if it was a long time ago I learned about it and my memory is hazy. Let me just point out that the phase diagram of the XY model is really weird, because there is no ordered phase below the KT transition.

The point is that domain walls destroy long-range order in low dimensions. In 1D a domain wall consists of a single pair of opposite spins, which can be excited at any non-zero temperature. In 2D you can only have long-range order in systems with discrete symmetry, like the Ising model. In systems with continuous symmetry, the energy of a domain wall decreases with its thickness. The cost of exciting a very thick domain wall is neglible, so again this will happen at any non-zero temperature.

However, the XY model is a bordering case, because below the KT transition the correlation length is infinite although there is no long-range order (this is called quasi-long-range order). Instead correlation functions decay with a power law, and the power depends on temperature. So there is a continuum of inequivalent critical systems up to the KT transition. Above that, the correlation length becomes finite.

Being 2D critical systems, this continuum is described by a continuum of inequivalent CFTs, all with c = 1.

• John Baez says:

That’s cool! I hadn’t heard of ‘quasi-long-range order’. I have the feeling that these technical subtleties about the XY model are part of why experts got so excited about the Kosterlitz–Thouless transition: it opened up new worlds that hadn’t been considered before.

14. John Baez says:

I’ve updated my blog article thanks to help from Greg Egan, Simon Burton, Simon Willerton and Haitao Zhang, and credited them for their work. It’s much better now!

15. There is some substantial information on the Nobel prize home page: https://www.nobelprize.org/nobel_prizes/physics/laureates/2016/advanced.html.

16. domenico says:

I have a problem near the phase transition; if the temperature is near the phase transition (for an infinitesimal difference in the temperature), and the material is homogeneus and without imperfections (so that the vortex-antivortex are uniform), then is there a temperature where the vortex-antivortex collide in a single singularity (in the form of a eight) for an overlap of vortices? I was thinking that a superconductive nanomaterial like a sphere, or a torus (without borders) could contain colliding singularities, or is it all too simple?

• John Baez says:

My first remark is that a phase transition can only occur in this kind of magnet—or indeed, almost any commonly studied physical system—when it has infinite spatial extent. Only in this limit do the expected values of physical quantities cease to be smooth functions of temperature! (For a large but finite system, these functions will be smooth, but some of their derivatives may become huge.)

So, a spherical or torus-shaped magnet of this sort won’t have a true phase transition. Nonetheless it’s very interesting to study vortices and antivortices on these topologies.

17. […] Empezamos por el “Premio Nobel de Física 2016: Los físicos teóricos Thouless, Haldane y Kosterlitz por los materiales topológicos 2D”, LCMF, 04 Oct 2016; puedes ver algunas animaciones de vórtices, antivórtices y de la transición de fase BKT en John Baez, “Kosterlitz–Thouless Transition”, Azimuth, 07 Oct 2016. […]

18. John Baez says:

This article got a bunch of hits from readers of the financial news magazine Bloomberg View, thanks to:

• Mark Buchanan, A Nobel that helps explain why a bagel always has hole, Bloomberg View, 13 October 2016.

The title and lead picture plays up the “topology as the science of bagels” theme that I was mocking, but the article gets past that in a nice way, while remaining utterly non-technical.

19. And Bob Dylan beats out Joan Baez for the Nobel Prize in Literature. Who’d a thunk?

20. […] But add even more energy and there is a critical level where the vortex and antivortex can separate. This is named the “Kosterlitz-Thouless transition” after two of the Nobel Prize awardees. It is a phase transition, meaning an abrupt change of state like the melting of ice into water at around 0°C or the evaporation of water into steam at around 100°C. (My summary is based on the very readable introduction by John Baez, who is also an internet relativity personality by the way.) […]

21. Robert Clark says:

Thanks for that well written explanation.

Bob Clark

22. André Oliveira says:

Dear John,

Let me ask a surely basic question (I’m not a physicist). I hope it even makes sense.

Take a plane without a point (which will be a vortex), so that its fundamental group is the integers.
Now, add an anti-vortex. It seems to me, from Brian Skinner’s diagram you posted above, that there cannot be loops around the anti-vortex, so anti-vortices do not change the topology of the plane independently if they are paired or not with a vortex.
By loop here I mean one induced by the spins of the atoms. One can only consider those, right?
From that diagram, it’s clear that the winding number of a loop around the vortex is 1, but I can’t see any loop around the anti-vortex, so how can one say the its winding number is -1?
Where am I wrong?

The basic question is: what is the topological invariant which gets changed when the KT phase transition occurs?
Thanks.

André.

• John Baez says:

André wrote:

The basic question is: what is the topological invariant which gets changed when the KT phase transition occurs?

None. No topological invariant changes when the KT phase transition occurs. If you read my description of what occurs, you’ll see it doesn’t mention any topological invariant changing. What changes is that the vortex-antivortex pairs become ‘unbound’. They are no longer stuck to each other. It’s a lot like how water breaks down into hydrogen and oxygen atoms when you heat it enough.

From that diagram, it’s clear that the winding number of a loop around the vortex is 1, but I can’t see any loop around the anti-vortex, so how can one say the its winding number is -1?

Trace your finger clockwise around the largest possible circle that fits in this picture:

See how the little arrow at the location of your fingertip rotates as you move your finger. It rotates counterclockwise one full turn.

Since it rotates one full turn in the opposite direction as the motion of your finger, we say this vector field has winding number -1, or more precisely ‘index’ -1. For more, try:

• Wikipedia, Index of a vector field.

• André says:

Thank you for your reply.
I now understand the picture on the winding numbers. The thing was that I was only considering curves integrating the vector field given by the spinors (I thought there was some physical reason for that). That’s why I wasn’t seeing any loop around an anti-vortex. But, now I understand.

Regarding the topological invariant, your answer (that there is no topological invariant changed when the KT-transition occurs) makes me confused. Does that mean that the topology of the thin films of magnetic material is the same before and after the KT-transition? If so, then what does all this has to do with topology? I thought that one of the main points of the Nobel on this prize was the “topological phase transitions”, hence the “bagel and pretzel” Nobel presentation, emphasising on topological invariants.

• John Baez says:

André wrote:

The thing was that I was only considering curves integrating the vector field given by the spinors.

‘Spinors’ don’t show up in the simple model we’re discussing, which is the one for which Kosterlitz and Thouless they won the Nobel prize. It’s called the classical XY model. The spin of each atom is modeled as a unit vector in the plane. There are also fancier quantum versions that involve spinors.

Does that mean that the topology of the thin films of magnetic material is the same before and after the KT-transition?

Of course the topology of the film remains the same: it’s typically a disk. Nobody is poking a hole in it when the phase transition occurs.

More interestingly, there is no topological invariant that suddenly changes when the phase transition occurs!

If so, then what does all this has to do with topology?

The ‘vortex number’ or index is a topological invariant of a vector field on the plane, which is 1 for a vortex, -1 for an antivortex, and m – n for a system with m vortices and n antivortices. In the lowest energy state the vortex number is 0, so the work of Kosterlitz and Thouless focuses on vector fields with vortex number 0. However, this allows for either a gas of vortex-antivortex pairs (which is what we get at low temperatures), or a gas of individual vortices and antivortices (which is what we see at higher temperatures, above the KT transition).

I thought that one of the main points of the Nobel on this prize was the “topological phase transitions”, hence the “bagel and pretzel” Nobel presentation, emphasising on topological invariants.

In my blog article here I made fun of those bagels and pretzels, because they don’t help anyone understand what Kosterlitz and Thouless won the Nobel prize for. They only help explain the scary word “topology”.

I should add, however, that Haldane also won the Nobel prize this year, and his work on 1-dimensional magnets (‘spin chains’) uses topology in a somewhat different way than Kosterlitz and Thouless did.

For more details on this, I really recommend the essay written by the Nobel prize committee:

• The Nobel Prize in Physics 2016,
Topological phase transitions and topological phases of matter.

23. André says:

Yes, the bagels/pretzels, instead of helping me, got me confused. I tried to read the Nobel essay, but it’s way too advanced for me. Thank you for your great explanations!

24. Thank you for this explanation! I would like to read the whole Nobel essay but I consider it’s a good introduction :)

25. It was the beginning of October the last time we met, and what happened next was a week plenty of exciting science: Nobel!!! Let’s learn (briefly) who won, plus (to make up for the delay), a list of associated curiosities.

26. VrilYa says:

Shouldn’t the antivortex be the same as vortex with opposite spin?﻿

• John Baez says:

No. I thought that too at first, but since you can smoothly interpolate between the ‘counterclockwise vortex’ and the ‘clockwise vortex’, they are two states of the same particle:

Furthermore, the clockwise and counterclockwise vortices can’t annihilate each other, but the true antivortex can annihilate the vortex:

• VrilYa says:

You are wrong there, in fluid mechanics two vortices with different spin direction do annihilate.

• John Baez says:

I’m not wrong; the vortices I’m talking about in this article are quite different than those in fluid dynamics.

• juvenwang says:

Can one show that the the clockwise (winding number 1) and counterclockwise (winding number 1) vortices (say, both winding number 1) can give rise a winding number 2 vortex when they merge?

• John Baez says:

That would be topologically allowed, but I believe that in this system there’s no difference between a “winding number 2 vortex” and two winding number 1 vortices that happen to be very close. Nobody seems to talk about “winding number 2 vortices” in this particular system.

27. mike4ty4 says:

I noticed immediately and instantaneously that the “antivortex” pattern looks interestingly like one for which the “field lines” look like hyperbolae preserved by a hyperbolic rotation (“Procrustean stretch” but also Lorentz transformation and split-complex multiplication). The duality of the vortex and antivortex then seems like it may be connected to and in the same broad bundle as that between Euclidean and Minkowski space, complex numbers and split-complex numbers, rotations through real and (in a sense) imaginary angles, and Riemannian and Lorentzian universes (cf. Greg Egan and “Orthogonal”) as well as “imaginary” speed of light and real speed of light :) Is there something neat and deep going on here?

28. […] اینجا هم خوب توضیح داده شده. […]

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