The Renormalisation Group

A new video which more or less completes the critical phenomena series. Jump straight to it if you want to skip the background.

One of my favourite topics is the critical point. I've posted many times on it, so to keep this short you can go back here for a summary. In brief, we're looking at a small point on the phase diagram where two phases begin to look the same. The correlation length diverges and all hell breaks loose. Well, lots of things diverge. At the critical point all length scales are equivalent and, perhaps most remarkably, microscopic details become almost irrelevant. Different materials fit into a small number of universality classes that share broad properties such as symmetry or dimensionality.

For a long time this universal nature was known about but it couldn't be said for sure if it was a truly universal thing, or just a really good approximation. Then along came the renormalisation group (RG), which gave a strong theoretical basis to critical phenomena and sorted everything out.

The renormalisation group is usually at the back end of an advanced statistical mechanics course, and that is not the level I'm going for with this blog. However, when making the videos for the demonstration of scale invariance and universality it became apparent that, even just making the pictures for these videos, I had to use RG. Even if I didn't realise it.

First I'll try to explain schematically what RG aims to do. Then I'll show how this is similar to how I make my pictures, and finally we'll get to a demonstration of RG flow at the end. I'll try not to dumb it down too much but I also want to be as quick as possible.

Renormalisation group

Let's look at how we do this with the Ising model. A simple model for a magnet where spins, sigma, (magnetic dipoles) can point up or down, $latex \sigma=\pm 1$, and like to align with their neighbours through a coupling constant, $latex J$. The energy is a sum over nearest neighbour pairs

$latex \displaystyle E=\sum_{ij} -J \sigma_i \sigma_j$

Where RG enters is to say that, if the physics is the same on all length scales, then we should be able to able to rescale our problem, to cast it on a different length scale, and get back the same thing. In real-space RG this is done by blocking. We bunch a group of our spins up together and form a new super spin that takes on the majority value of its constituents. It's as though the spins in the block get together and vote on how they want to be represented, and then we can deal with them as one.

Here's what it looks like. Take an Ising model configuration

Block them together

And vote

We're left with a pixelated version of the first picture. Now here I will slightly deviate from RG as standard. The next step is to ask, if these super spins were a standalone Ising model, what temperature would they have? If our initial system is right on the critical point then the renormalised (blocked) system should have the same temperature because it should look exactly the same – scale invariance. If you're even slightly off then the apparent temperature, let's call it $latex T_{RG}$, will flow away from the critical point towards a fixed point.

These fixed points are the ferromagnet (all spins the same, $latex T_{RG}=0$) or the paramagnet (completely random, $latex T_{RG} \rightarrow \infty$) as shown below.

Normally RG is done in terms of coupling constants rather than temperature. However, I think in our case temperature is more intuitive.

Zooming out

By now the link between RG and the pictures I make may already be clear. The configurations I will show below are made of something like $latex 10^{10}$ spins. Clearly I can't make a 10 Giga pixel jpeg so I have to compress the data. In fact the way I do it is an almost identical blocking process. Spins are bundled into $latex b \times b$ blocks and I use a contrasting function (a fairly sharp tanh) that is not far away at all from majority rule as described above.

If we start by zooming in to a 768x768 subsection then each pixel is precisely one spin. As we zoom out we eventually need to start blocking spins together. In the video below there are three systems: one ever-so-slightly below $latex T_c$, one ever-so-slightly above $latex T_c$ and one right on the money. At maximum zoom they all look pretty much the same. If you had to guess their temperatures you'd say they're all critical.

As we start to zoom out we can see structure on  length scales, and the apparent temperatures start to change, in fact they flow towards the fixed point phases. Video below, recommend you switch on HD and watch it full screen.



So there it is. RG in action. If you're not precisely on the critical point then you will eventually find a length scale where you clearly have a ferromagnet or a paramagnet. At the critical point itself you can zoom out forever and it will always look the same. The renormalisation group is a really difficult subject, but I hope this visualisation can at least give a feeling for what's going on, even if the mathematical detail is a bit more challenging.

Universality at the critical point

Time for more critical phenomena.

Another critical intro

I've talked about this a lot before so I will only very quickly go back over it. The phase transitions you're probably used to are water boiling to steam or freezing to ice. Now water is, symmetrically, very different from ice. So to go from one to the other you need to start building an interface and then slowly grow your new phase (crystal growth). This is called a first order phase transition and it's the only way to make ice.

Now water and steam are, symmetrically, the same. At most pressures the transition still goes the same way – build an interface and grow. However, if you crank up the pressure enough there comes a special point where the distinction between the two phases becomes a bit fuzzy. The cost of building an interface goes to zero so there's no need to grow anything. You just smoothly change between the two. This is a second order, or continuous, phase transition and it's what I mean by a critical point.

As I've demonstrated before, one of the consequences of criticality is a loss of a sense of scale. This is why, for instance, a critical fluid looks cloudy. Light is being scattered by structure at every scale. This insight is embodied in the theory of the renormalisation group, and it got lots of people prizes.

Universality

A second feature of critical phenomena is universality. Close to the critical point it turns out that the physics of a system doesn't depend on the exact details of what the little pieces are doing, but only on broad characteristics such as dimension, symmetry or whether the interaction is long or short ranged. Two systems that share these properties are in the same universality class and will behave identically around the critical point.

At this stage you may not have a good picture in your head of what I mean, it does sound a bit funny. So I've made a movie to demonstrate the point. The movie shows two systems at criticality. On the left will be an Ising model for a magnet. Each site can be up or down (north or south) and neighbouring sites like to line up. The two phases at the critical point are the opposite magnetisations represented here by black and white squares.

On the right will be a Lennard-Jones fluid. This is a model for how simple atoms like Argon interact. Atoms are attracted to one another at close enough range but a strong repulsion prevents overlap. The two phases in this case are a dense liquid and a sparse gas.

One of these systems lives on a lattice, the other is particles in a continuous space that are free to move around. Very different as you can see from the pictures. However, what happens when we look on a slightly bigger length scale? Role the tape!


At the end of the movie (which you can view HD) the scale is about a thousand particle diameters across containing about 350,000 particles and similar for the magnet. At this distance you just can't tell which is which. This demands an important point: These pictures I've been making don't just show a critical Ising model, they pretty much show you what any two-dimensional critical system looks like (isotropic, short range...). Even something complicated from outside of theory land. And this is why the theory of critical phenomena is so powerful, something that works for the simplest model we can think of applies exactly - not approximately - to real life atoms and molecules, or whatever's around the kitchen.

Critical Point

I'm finally getting around to sharing what, for me, is the most beautiful piece of physics we have yet stumbled upon. This is the physics of the critical point. It doesn't involve enormous particle accelerators and it's introduction can border on the mundane. Once the consequences of critical behaviour are understood it becomes truly awe inspiring. First, to get everyone on the same page, I must start with the mundane - please stick with it, there's a really cool movie at the bottom...

Most people are quite familiar with the standard types of phase transition. Water freezes to ice, boils to water vapour and so on. Taking the liquid to gas transition, if you switch on your kettle at atmospheric pressure then when the temperature passes 100 degrees centigrade all the liquid boils. If you did this again at a higher pressure then the boiling point would be at a higher temperature - and the gas produced at a higher density. If you keep pushing up the pressure the boiling point goes higher and higher and the difference in density between the gas and the liquid becomes smaller and smaller. At a certain point, the critical point, that difference goes to zero and for any higher pressure/temperature the distinction between the liquid and gas becomes meaningless, you can only call it a fluid.

The picture below, taken from here, shows the standard phase diagram, with the critical point marked, for water.




Magnets also have a critical point. Above the critical temperature all the little magnetic dipoles inside the material are pointing in different directions and the net magnetisation is zero. Below the critical temperature they can line up all the in the same direction and create a powerful magnet. While the details of this transition are different from the liquid-gas case, it turns out that close to the critical point the details do not matter. The physics of the magnet and the liquid (and many other systems I won't mention) are identical. I'll now try to demonstrate how that can be true.

The pictures below are taken from a computer simulation of an Ising model. The Ising model is a simple model for a magnet. It's been used for so much more than that since its invention but I don't really want to get into it now. For the pictures below squares are coloured white or black. In the Ising model squares can change their shade at any time, white squares like to be next to white squares and black squares like to be next to black squares. Fighting against this is temperature, when there is a high temperature then squares are happier to be next to squares of a different colour. Above the critical temperature, if you could zoom out enough, the picture would just look grey (see T=3 below). Grey, in terms of a magnet, would be zero magnetisation.







If you drop the temperature then gradually larger and larger regions start to become the same colour. At a certain point, the critical point, the size of these regions diverges. Any colder and the system will become mostly white, or mostly black (as above, T=2). Precisely at the critical point (T=2.69 in these units), however, a rather beautiful thing happens. As the size of the cooperative regions diverge, so too do fluctuations. In fact at the critical point there is no sense of a length scale. If you are struggling to understand what this means then look at the four pictures below. They are snapshots of the Ising model, around the critical point, at four very different scales - see if you can guess which one is which.








Now watch this movie for the answer (recommend switching to HD and going full screen).





The full picture has 2^34 sites (little squares), that's about 17 billion. This kind of scale invariance is a bit like the fractals you get in mathematics (Mandelbrot set etc) except that this is not deterministic, it is a statistical distribution.

How does it demonstrate that the details of our system (particles, magnetic spins, voting intentions - whatever) are not important? In all these cases the interactions are short ranged and the symmetry and dimension are the same. Now imagine that you have a picture of your system (like above) at the critical point and you just keep zooming out. After a while you'll be so far away that you can't tell if it's particles or zebras interacting at the bottom as that level of detail has been coarse grained out and all the pictures look the same. This is not a rigorous proof, I just want to convey that it's sensible.

Of course the details will come into play at some point, the exact transition temperature is system dependent for example, but the important physics is identical. This is what's known as universality, and it's discovery, in my opinion, is one of the landmarks in modern physics. It means I can take information from a magnet and make sensible comments about a neural network or a complex colloidal liquid. It means that simple models like the Ising model can make exact predictions for real materials.

So there it is. If you don't get it then leave a comment. If you're a physics lecturer and you want to use any of these pictures then feel free. I'd only ask that you let me know as, well, I'd like to know if people think it's useful for teaching. For now you'd have to leave a comment as I haven't sorted out a spam-free email address.

UPDATE: Forward link to a post on universality.