# Reasons for violation of universality in statistical mechanics

The Universality in statistical mechanics is nicely explained by the renormalization group theory. However, there are fair amount of numerical and theoretical studies show that it can be violated in models such as Ising model, spin glass, polymer chain and percolation (Example references: 1, 2, 3, 4, 5, 6, 7) and some violations are pretty strong. So, my questions are:

• What are the general reasons of the violation of the universality?
• Also, why can't they be explained by the renormalization group theory and scaling hypothesis?
• Is it always true that there are some "core elements" for each universality class? Whenever they are missing, we can definite say they are not in the same universality class.
• Is there an example of commonly recognized "universality class" (such as 2D Ising model) that show different critical exponents by changing the microscopic details? Note, it should not happen by the definition of universality class, but there is no proof for all universality class.

I would accept answers discussing solid examples for some universality class about these questions.

Note: Universality refers to a large class of different systems exhibiting the same properties regardless of its microscopic details such as the lattice type (square, hexagonal, triangular, and kagome lattices). In statistical mechanics, the universality class are usually determined by the symmetry of the order parameter (such as up-down symmetry in Ising model and spherical symmetry in Heisenberg model) and the topological structure (such as dimensionality).

• No one can tell me some answers of these questions? At least I want some evidents of why or why not these questions are valid or not. Jan 7, 2013 at 4:50

## 1 Answer

Your reference $3$, for instance, is an Ising model modified by randomness. It is of course not clear how universality works in such complex systems. Another link studies (finite size) clusters, far from the ideal infinite system of the analytical solution.

In general, recall that universality is a feature of phase transitions with respect to a parameter of the system. Models are typically constructed using a lattice Hamiltonian and phase transitions derived from the partition function, with $1/T$ as the crucial parameter. The (solved) 2D Ising model illustrates the importance of conformal symmetry and related field theories for typical critical phenomena. But few systems are exactly solved, and systems may be studied away from criticality.

Renormalisation a la Wilson turns the critical point into fixed points for RG flows. Again, the process only works for relatively simple Hamiltonians. (Such techniques are now used even in the 4D case)!

• So do you mean that there is no general explanation and indicators for the violation? Is there anything that we can say that the universality is not valid anymore when a particular feature in the model is found? Say the randomness you discussed. There should be models show scaling that differ from the prediction of renormalization, but still exhibit universality. It seems that I read similar stuff, but I am not quite sure about that though. Also, I dont think there is any problem for the finite size system, they have preform scaling anaylsis and the conclusion seems still valid. Jan 6, 2013 at 9:54
• The term 'universality' relates to solvable systems with the parameters belonging to a specific class. Anyway there is no generic one size fits all answer to the breaking of universality - because in principle we could solve a complicated system one day and discover new universality classes. The concept of universality hinges only on the independence of the parameters from all physical details of the system. Yes it is more general than renormalization itself. And we cannot definitively say that universality is 'broken' just because we do not recognize the features of a given transition. Jan 6, 2013 at 21:18
• I dont believe that there is a one size fit all anwsers since each universality class are intrinsically different. However, I still feel that we can say something on it. If a property is varied, and the transition change from second order to first order, then we can say something. Universality is interesting not only in theory. Even the model is not solvable, the critical exponents could still be figured out by experiment in which the experimental details are not important such as structural deformation. Jan 7, 2013 at 4:38
• I agree, yes, the experimental point of view is also interesting. Jan 7, 2013 at 21:15