conformal field theory - Universality classes

I would like to ask about the universality classes. I know that these are the statistical models which describes different phase transitions with different critical exponents. But I would like to know what is the difference of the universality classes and what is exactly an universality class? Could someone help me to summarize the models and tell me the story of the models, how came to be following the idea that the Ising-model shows a scale invariance in the critical point?

Answer

A universality class is an equivalence class of physical models – field theories, quantum field theories, or models of classical or quantum statistical physics – where the equivalence is defined by two or several models' having the same mathematical description of the behavior at very long time scales and distance scales.

So if two models' behavior at very long scales is described by the same mathematics with the same parameters etc., they belong to the same universality class. In some mathematical sense, they are the same even though they may describe very different materials in different conditions or even very different subdisciplines of physics.

This concept is a powerful concept because the number of "universality classes" is rather limited. The equations describing the limit of the very long scales has to be a scale-invariant theory and there aren't too many. If one studies the long-distance behavior of many models, one simply gets an empty set, the most uninteresting universality class. Some universality classes contain mutually non-interacting excitations; they are not too interesting, either.

Interacting scale-invariant theories are rather rare. The Ising model (at the critical point) is an excellent example. This conformal field theory is used by condensed matter physicists as well as e.g. string theorists. Those subdisciplines find this nontrivial behavior in physical systems of very different composition. The Ising model in string theory describes some degrees of freedom living on a relativistic string that may describe a part of the string's vibrations in the compactified dimensions, for example! But the mathematics is the same.

This universality class – and similar interacting classes – are very interesting. Something is happening at all length scales and time scales. Why? Because we assumed that something is happening – it is a nontrivial, interacting theory. But because it is scale-invariant, it doesn't have a preferred length scale or time scale. So if there are "blobs" of a certain size that appear and disappear in the system, it must be blobs of all possible sizes, at all possible time scales. (Well, the blobs really exist only at length scales much longer than the atomic radius. That's because we first had to take the long-distance limit which allowed us to eliminate the non-scale-invariant behavior at the atomic radius scale or shorter scales.)

The undergraduate way to think of the critical phenomena, scale-invariant (or conformal) theories, and universality classes is e.g. critical opalescence. Near the critical point, one sees very complicated and dynamical fog which has no preferred characteristic length scale or time scale. If you magnify it (and scale the time in a corresponding way), you get pretty much the same picture as you had before the scaling.