# List of known universality classes

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I am working with RG and have a pretty good idea of how it works. However I have noticed that even though the idea of universality class is very general and makes it possible to classify critical systems, textbooks seem to always end up with the Ising model as an example. As a consequence my knowledge of other universality classes is very poor.

My question is simple: What other universality classes are there and what are their properties (in particular, the value of their critical exponents)?

I know that there are as many universality classes as there are RG fixed points and that my question can never be answered completely. A list of 4 or 5 (thermal equilibrium) universality classes that are well established and understood would however give me the feeling that there is more than Ising model out there.

I will of course very much welcome references to literature. The reviews that I know on RG usually focus on general aspects and give few examples.

This post imported from StackExchange Physics at 2014-10-09 19:55 (UTC), posted by SE-user Steven Mathey
retagged Oct 9, 2014
Perhaps the question could be phrased something like "is there a well-known classification of universality classes for (insert relevant integer here)-dimensional field theories?" If so, is it possible to write this classification in a succinct way? What does that look like?" I think that in a form that is something like this, it's a very useful conceptual question.

This post imported from StackExchange Physics at 2014-10-09 19:55 (UTC), posted by SE-user joshphysics
Universality classes are classified by space dimensionality and realised symmetries. That is what the textbooks say before they go to Ising model. I'm asking about particular examples.

This post imported from StackExchange Physics at 2014-10-09 19:55 (UTC), posted by SE-user Steven Mathey

Any CFT can be a RG fixed point. So for instance, in 2D, all the minimal models may be viewed as universality classes of which the 2D Ising would be a particular example.

Could one say that there is same kind of a (one to one?) correspondance between fixed point and a universality class? Can a universality class generally be discribed/characterized by a certain fixed point?

Yes, I would say that every distinct fixed point is a universality class.

''every distinct fixed point is a universality class'' doesn't make sense as different theories will have different fixed points but may belong to the same universality class. A universality class is an equivalence class of fixed points with the same local asymptotic behavior. It is the latter that characterizes the universality class.

A fixed point, which simplistically I take to be a CFT, is uniquely determined by things like central charge, spectrum of primaries, and so on. I don't particular pay emphasis on how the CFT is reached which is the equivalence class that Neumaier mentions. That is the basis of my statement "Every distinct fixed point is a universality class."

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