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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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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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