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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?

I know there are as many universality classes as there are RG fixed points, so my question can never be answered completely. A list of 4 or 5 (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.

knzhou
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Steven Mathey
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  • Big list type questions are discouraged on this site. You might want to try narrowing your focus. – Kyle Kanos Oct 08 '14 at 16:24
  • That's why I added the last paragraph. If I knew how to narrow it down I wouldn't be asking the question. – Steven Mathey Oct 08 '14 at 16:32
  • The phrase "most important" is also an opinion based one (someone doing X would agree X is best while someone doing Y would agree Y is best). – Kyle Kanos Oct 08 '14 at 16:36
  • This meta post about big-list questions might be relevant here. – Ali Oct 08 '14 at 17:02
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    @Ali I edited your comment to link to the original question, so as to give people the proper context. – David Z Oct 08 '14 at 17:47
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    Also, I would note that the problem is really with questions that encourage the "one answer per post" style of responding. If someone were to post an answer giving the entire list, all together in one place, that would be fine. And I think it's possible to edit this question in a way that prompts that. Thoughts? – David Z Oct 08 '14 at 17:49
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    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. – joshphysics Oct 08 '14 at 18:00
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    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. – Steven Mathey Oct 08 '14 at 18:19
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    @StevenMathey Right: I'm trying to help you out here so that this question doesn't get closed. – joshphysics Oct 08 '14 at 21:57
  • I'm sorry, I don't think that I get the problem. I know that I'm asking for a lot, but a partial answer would already be great. If 2 or 3 people describe their favourite universality class I will be satisfied. – Steven Mathey Oct 08 '14 at 22:07
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    I stumbled on this today. It's not complete, but is a very good start. – Steven Mathey Oct 12 '14 at 00:39
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    We do have a rather comprehensive idea of a variety of static universality classes for equilibrium systems, ranging from the SAW to an n-component nonlinear sigma model (all in different dimensions), but the exponents fro a lot of these are still speculative and conjectured (heuristically or through RG or numerics or some other method). The issue as others have pointed out is the lack of exactly solvable models to characterize non-trivial universality classes. Classification of dynamical universality classes on the other hand is far richer and much less is known about them. – surajshankar Oct 12 '14 at 03:26
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    You might know them already, but a couple references that I know are "Lectures On Phase Transitions And The Renormalization Group" by Nigel Goldenfeld and "Critical Dynamics" by Uwe Tauber. Hope they help! – Ben Mar 27 '15 at 17:40
  • Before mindlessly thrashing around in the Domb & Green collection, I *strongly* recommend mastering the UC coverage of Jean Zinn-Justin's classic sourcebook, Quantum Field Theory and Critical Phenomena (International Series of Monographs on Physics) 4th Edition, International Series of Monographs on Physics (Book 113), Clarendon Press; 4 edition (August 15, 2002) ISBN-10: 0198509235 – Cosmas Zachos Feb 04 '16 at 16:47

2 Answers2

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Two systems belonging to the same universality class will have the same critical exponents.

There are many things that determine the universality class of a system, one being its dimension.

The 2D Ising model is one of the most studied system in statistical mechanics because it admits an exact soultion, found by Lars Onsager in 1944. Its critical exponents are:

$$\alpha = 0 \ \ \ \beta = 1/8 \ \ \ \gamma = 7/4 \ \ \ \delta = 15 \ \ \ \nu = 1 \ \ \ \eta= 1/4$$

But let's take the (experimental) values of the critical exponents for the 3D Ising model:

$$\alpha = 0.110 \ \ \ \beta = 0.327 \ \ \ \gamma = 1.24 \ \ \ \delta = 4.79 \ \ \ \nu = 0.630 \ \ \ \eta= 0.0364$$

So the 3D Ising model belongs to a different universality class. Or we can take 2D percolation (which is exactly solvable):

$$\alpha = -2/3 \ \ \ \beta = 5/36 \ \ \ \gamma = 43/18 \ \ \ \delta = 91/5 \ \ \ \nu = 4/3 \ \ \ \eta= 5/24$$

So another universality class. Other universality classes will be for example that of 3D percolation, the Heisenberg model or the Van der Waals gas. Here is a list.

I conclude by saying that every system has an upper critical dimension (es D=4 for the Ising model and D=6 for percolation), above which the critical exponents become constant and can be computed using mean-field theory. The mean-field values of the critical exponents are:

$$\alpha = 0 \ \ \ \beta = 1/2 \ \ \ \gamma = 1 \ \ \ \delta = 3 \ \ \ \nu = 1/2 \ \ \ \eta= 0$$

These values are the same as the ones of the Van der Waals gas; so the VdW gas, the $4(5,6,7...)$-D Ising model and the $6(7,8,9...)$-D percolation are examples of systems belonging to the same universality class: the mean field class.

valerio
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3

Other examples are

The Ashkin-Teller and Potts models can be mapped onto the 8-vertex model. The latter is then mapped onto the Coulomb gas whose critical properties are known from RG.

Christophe
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