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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).

unsym
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1 Answers1

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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)!

BrianWa
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