# Is there a list of all universality classes for phase transitions with examples of each?

I've often had this problem: I have a model that has a phase transition in it, but I don't know what universality class it falls into or what the universality class is called.

Is there anywhere on the web where there is a big list of all the classes and examples of each (hopefully with critical exponents as well)?

For example:
Name: 2D Ising Model
Symmetry: Z$$_2$$
Dimension: 2
Other examples: liquid-gas transition, etc
Critical exponents: ...

• Not quite an answer, but this paper gives a number of examples. I'm not too well versed in universality class so I don't know for sure if this is what you had in mind.
– Tyberius
May 1, 2020 at 22:01
• The 2D Ising model, along with Hubbard model, Anderson model, BCS model, XY model, Jaynes-Cummings model, tight binding model, and others, are given in the Hamiltonian Zoo, which is similar to the famous "Particle Zoo" but for Hamiltonians, and open-source so that anyone can add missing Hamiltonians. However the properties of these models (such as Z$_2$ symmetry, critical exponents, etc.) are not included. It would be really helpful if there was a website that had that information too! May 3, 2020 at 22:22
• Another interesting field to include in this list would be whether its properties are known exactly or only estimated, as well as a couple of main references. May 19, 2020 at 1:19
• Related, in Physics SE: Examples of important known universality classes besides Ising. May 19, 2020 at 1:24
• @NikeDattani Done (and I can't believe how long it took to essentially transcribe the table). Jun 13, 2020 at 3:03

A locally interacting system displaying a continuous phase transition belongs to a universality class that is determined solely by the system symmetries and dimensionality.

Drawing from Wikipedia's list (itself mostly based on Ódor's paper) and this answer from Physics SE, here's a partial list of universality classes and critical exponents:

$$\begin{array}{| c | c | c c c c c c | c|} \hline \textbf{dim.} & \textbf{Symm.} &\alpha & \beta & \gamma & \delta & \nu & \eta & \textbf{class} \\ \hline \hline \text{any} & \text{any} & 0 & 1/2 & 1 & 3 & 1/2 & 0 & \text{Mean field} \\\hline 2 & \text{Sym}_{2} & 0 & 1/8 & 7/4 & 15 & 1 & 1/4 & \\ 3 & \text{Sym}_{2} & 0.11007(7) & 0.32653(10) & 1.2373(2) & 4.7893(8) & 0.63012(16) & 0.03639(15) & \text{Ising} \\ 4+ & \text{Sym}_{2} & 0 & 1/2 & 1 & 3 & 1/2 & 0 & \\\hline 2 & \text{Sym}_{3} & 1/3 & 1/9 & 13/9 & & 5/6 & & \text{3-state Potts} \\\hline 2 & \text{Sym}_{4} & 2/3 & 1/12 & 7/6 & & 2/3 & & \text{Ashkin-Teller (4-state Potts)} \\\hline 3 & {\mathcal {O}}(2) & −0.0146(8) & 0.3485(2) & 1.3177(5) & 4.780(2) & 0.67155(27) & 0.0380(4) & \text{XY} \\ \hline 3 & {\mathcal {O}}(3) & −0.12(1) & 0.366(2) & 1.395(5) & & 0.707(3) & 0.035(2) & \text{Heisenberg} \\\hline 1 & \mathbf{1} & & 0 & 1 & & 1 & & \\ 2 & \mathbf{1} & −2/3 & 5/36 & 43/18 & 91/5 & 4/3 & 5/24 \\ 3 & \mathbf{1} & −0.625(3) & 0.4181(8) & 1.793(3) & 5.29(6) & 0.87619(12) & 0.46(8) \text{ or } 0.59(9) & \text{Ordinary percolation} \\ 4 & \mathbf{1} & −0.756(40) & 0.657(9) & 1.422(16) & & 0.689(10) & −0.0944(28) \\ 5 & \mathbf{1} & & 0.830(10) & 1.185(5) & & 0.569(5) & \\ 6+ & \mathbf{1} & −1 & 1 & 1 & 2 & 1/2 & 0 \\ \hline 1 & \mathbf{1} & 0.159464(6) & 0.276486(8) & 2.277730(5) & 0.159464(6) & 1.096854(4) & 0.313686(8) & \\ 2 & \mathbf{1} & 0.451 & 0.536(3) & 1.60 & 0.451 & 0.733(8) & 0.230 & \text{Directed percolation}\\ 3 & \mathbf{1} & 0.73 & 0.813(9) & 1.25 & 0.73 & 0.584(5) & 0.12 \\ 4+ & \mathbf{1} & −1 & 1 & 1 & 2 & 1/2 & 0 \\ \hline \end{array}$$

where $$\mathbf{1}$$ denotes the trivial group, $$\text{Sym}_{n}$$ the $$n$$-th Symmetric group, and $$\mathcal{O}(n)$$ the Orthogonal group.

• +100. This is a heroic effort !!!! Jun 13, 2020 at 3:20
• @NikeDattani That's very generous. Reformatting was not without toil, true, but the information was compiled by Ódor in this paper and transcribed to Wikipedia by a bunch a people, so my own contribution is modest in comparison. Jun 13, 2020 at 3:33
• So you were standing on the shoulders of giants! Do you think we should cite the Ódor paper here? Jun 13, 2020 at 3:38
• @NikeDattani Sure, I had meant to but forgotten. Jun 13, 2020 at 10:18
• This is very nice. But since these are universality classes and the point is that they describe multiple critical points would it be possible to add multiple examples of realizations for each class? That would be great! Jul 15, 2020 at 21:47

I felt that it's important to note that not all universality classes are determined solely by spacial dimension and symmetry, even when the interactions are local.

The easiest model that exemplifies this is probably the Ashkin-Teller model in 2D. It has a continuously varying (thus infinite variety of) critical exponents (universality classes) depending on the value of the parameter in the Hamiltonian. The symmetry does not change for different values of the parameter, so this is a counter example to the often made claim that universality classes are determined only by dimensionality and symmetry. The critical exponent values written in the wikipedia chart with "Ashkin-Teller" only corresponds to ONE point of this continuum of criticalities.

Of course, this kind of examples are rare, and usually universality classes could indeed be predicted from the spacial dimension and the symmetry. But I think it's important to realize that the claim is not a rigorously proven theorem, but is more like a guideline. There are even recent works on trying to find a non-Ising $$Z_2$$ symmetry breaking universality class (https://arxiv.org/abs/1803.00578).

Also, to add to the list, I think the cubic-anisotropic Heisenberg universality class is one that is easy to be mistaken with the normal isotropic Heisenberg universality class. The critical exponents are close anyway, and it's actually hard to numerically distinguish them, but RG calculations say that the anisotropy perturbation is relevant, meaning that it flows to a separate fixed point.