Wednesday, 13 April 2011

Paper review: Hexatic phases in 2D

I'm doing my journal club on this paper by Etienne Bernard and Werner Krauth at ENS in Paris:

First-order liquid-hexatic phase transition in hard disks

So I thought that instead of making pen-and-paper notes I'd make them here so that you, my huge following, can join in. If you want we can do it proper journal club style in the comments. For now, here's my piece.

Phase transitions in 2D

Dimension two is the lowest dimension we see phase transitions. In one dimension there just aren't enough connections between the different particles – or spins, or whatever we have – to build up the necessary correlations to beat temperature. In three dimensions there are loads of paths between A and B and the correlations really get going. We get crisp phase transitions and materials will readily gain long range order. Interestingly, while it should be easier and easier to form crystals in higher dimensions there do exist pesky glass transitions that get worse with increasing dimension. But I digress.

In two dimensions slightly strange things can happen. For one thing, while we can build nice crystals they are never quite as good as the ones you can get in 3D. What do I mean by this? Well in 3D I can give you the position of one particle and then the direction of the lattice vectors and you can predict exactly where every particle in the box will sit (save a bit of thermal wiggling). In 2D we get close, if I give you the position and lattice vectors then that defines the relative position and orientation for a long way – but not everywhere.

By "a long way" I mean correlations decay algebraically (distance to the power something) rather than exponentially (something to the power distance), which would be short ranged. We can call it quasi-long ranged.

Never-the-less, this defines a solid phase and this solid can melt into a liquid (no long range order of any kind). What is very interesting in two dimensions is that this appears to happen in two stages. First the solid loses its positional order, then it loses it's orientational order as well. This is vividly demonstrated in Fig 3. of the paper. The phase in the middle, with quasi-long range orientational order but short range positional order, is known as the hexatic phase.

When the lattice is shifted a bit the orientation can be maintained but the positions become disordered.

A brief XY interlude

Before we get on to hard disks it might help to understand a slightly simpler model. The XY model is similar to the Ising model, it's on a lattice but the spins are now continuous in the XY plane. This is basically enough to kill the phase transition we see in the Ising model (separation of up and down spins) because the XY spins can gradually rotate from up to down getting rid of a sharp interface.

So we lose any long range order, however, at low temperatures the XY model can hold quasi-long range order – just like the hexatic. Most importantly there is a phase transition from disordered to quasi-ordered. This transition, known as the Kosterlitz–Thouless (KT) transition, is a bit weird and it is related to topological defects in the vector field. Chapter 9 of Chaikin and Lubensky will blow your head off if you want to learn all there is to know about these things. To get a rough idea here is what these defects, or "vortices" can look like (nicked from here).

These vortices cost free energy but at high enough temperature we can afford them. In turn they have affect of suppressing correlations in the director field. When the temperature drops such that we can't afford these defects, we develop quasi-long ranged order. The KT transition is continuous in nature (ie, you don't get interfaces) although it's not strictly second order.

Back to disks

Going back to particles, this hexatic picture appears to be fairly ubiquitous in 2D systems. To get something more concrete we now focus on one model, the simplest of all the off-lattice models, hard disks. Hard disks, like hard spheres, are very interesting because they are the basis of most liquid theories and are the simplest approximation to an atom we can think of. So how do they melt?

No one pretends (or pretended) to know for sure. Most people agree that there should be a (quasi) solid that melts to a liquid via a hexatic but the nature of the transition is hotly contested. One prediction is via KTHNY theory. This horribly named theory (I now pronounce either "kuthny" or I cough and wave my hands) can give two continuous, XY-esque, transitions: solid-[continuous]->hexatic-[continuous]-> liquid.

So we finally arrive at the paper. What Etienne and Werner are now saying is that the transition from liquid to hexatic is actually first order. The reason it is so difficult to say for sure is that you need a very big system to see it and, because it's so dense, you need a long time to equilibrate it. In this paper they have a huge system (10^6 particles) and they use a special Monte Carlo algorithm, the Event Chain algorithm, that is very efficient for hard disks. These together allow them to really see what the transition looks like.

To verify what they're seeing they first study the two phases in isolation, just above and below the coexistence region. By monitoring the pressure as they scan the coexistence region (in an infinite system it would be constant) they can see how the peak-to-peak pressure difference scales. The scaling is quite clean and consistent with a first order transition. The most vivid demonstration of the first order nature is the picture in Fig 1. that shows the interface between the liquid and the hexatic phase.

First order then?

So the journal club part of this is to ask how convinced you are? My brain is naturally attracted to pictures and that interface is pretty striking. I happen to know they've done simulations with 4x the area and it looks even better there. As it's shown here there's maybe a bit of ambiguity just by eye.

If you want to remember what a second order transition looks like there's always the super huge Ising model (now in HD!).

The pressure measurements are probably the most convincing piece of evidence for me. It's certainly an impressive achievement, I definitely look forward to any follow-ups.


  1. how does the pressure measurement related to the kinds of phase transitions?

  2. Hi. Been away for a while and just spotted that blogger has quarantined a bunch of comments.

    The pressure measurements show rather nice Van de Waals loops and the scaling fits that of a first order phase transition rather than a continuous one. So it's how the pressure measurements scale with system size that is interesting.

    Hope I've got that right.