# Things I don’t understand

What are the different possible phases of matter?

One of the big advances of twentieth century physics was the development of a very general set of ideas – the renormalization group – that let you analyse and understand the properties of different phases of matter, and the phase transitions between them. This development was done by a whole bunch of people, including Landau, Ginzburg, Kadanoff, Michael Fisher, Wilson, and others.

In a recent issue of *Science* there’s an article by Senthil, Balents, Sachdev, Vishwanath and Matthew Fisher claiming to have found a significant class of phase transitions that can’t be understood within this framework. (Here’s the long (and possibly more comprehensible) version of the paper at a publicly accessible site.)

This would seem to be extremely significant if true, which is why I’m reading the paper. I’m reminded, as I read, however, of the many basic items of background material I don’t understand all that well.

One thing that always bugs me when I read about phase transitions is the question “What is an order parameter?” Landau introduced this concept as the unifying idea behind his theory of phase transitions. Examples include the magnetisation of a ferromagnet, and the phase in a superconductor.

So far as I can tell, the order parameter is usually divined, as opposed to defined. How are we supposed to deduce the order parameter? Is there a freedom in our choice of order parameter? What makes one choice of order parameter a good one? I’d love to fully understand the answers to any of these questions.

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Order parameters rock. OK, now that I’ve gotten that out of my system… To answer one of your questions:

There are often many order parameters for a phase transition. Consider, for example, the 2D Ising model. One order parameter for the system is the magnetization, like you said. Thus if we take the state of our Ising model, add up all the up spins and subtract all the down spins, we arrive at the total magnetization. At low temperatures the system will be in one of two ordered states and the magnetization will be in one of the two bifurcated state plus or minus M. Raise the temperature and eventually the system will undergo a phase transition and the magnetization will vanish.

But you can also take a more complicated order parameter which also shows the phase transition. The ground state of the Ising model is a redudancy code. As excitations from the ground state occur, they violate Ising bonds with an energy proportional to the perimeter of the flipped spins. So at any non zero temperature, the system will be in a state where there are a lot of small domains of flipped spins. An order parameter can be constructed using this fact. Take the state of your spins. Now on the dual lattice construct occupied edges if the corresponding Ising bond is violated (spins antialigned.) These occupied edges will form closed strings. An order parameter can then be constructed where an iterative process of shrinking these domains is applied and the final order paramter is the final state of the final domain.

Of course the reason why the order parameter magnetization is so much of a better order parameter is that it is easily measurable. The order paramter I described is much harder to measure (you effectively do such a measurement if you quickly ramp you temperature to super close to T=0, let the system relax, and then measure the magnetism.)

Dave: It’s still not clear to me what the order parameter actually is, in any given situation.

Here’s the little I can divine:

* Order parameters are observables. (I’m not 100% sure your examples are, though.)

* The observable is usually traceless, ensuring that at infinite temperature, the average of the order parameter vanishes. At high temperature the order parameter is still zero, on average, but becomes non-zero below some transition temperature.

* Otherwise, I’m not sure what restrictions we can place.

The shrink island order parameter is indeed an observable: it gives a one to one mapping between the states of the spins and a value (+1 or -1.) The point is that it differs sometimes from the total magnetization (which is an observable spread between +1 and -1 (or +M and -M, whatever)) But it shouldn’t differ for the location of the phase transition.

I don’t know how much being traceless matters. It just shifts things around. So we can have an order parameter of S at high temperature and at lower temp it bifurcates in S(+/-)M.

Order parameter, to me, is just a fancy way of saying, (1) this is the property of the system we are interested in and (2) look this property indicates that there are multiple phases in the system. Constructing order parameters is then all about the intution behind why such a property is important in reveal “what” the system “is” [it may just be that it’s easily measurable, also. but all the good order parameters, like those you find in High Tc superconductors look harder than heck to measure.]