Universality class


In statistical mechanics, a universality class is a collection of mathematical models which share a single scale invariant limit under the process of renormalization group flow. While the models within a class may differ dramatically at finite scales, their behavior will become increasingly similar as the limit scale is approached. In particular, asymptotic phenomena such as critical exponents will be the same for all models in the class.
Some well-studied universality classes are the ones containing the Ising model or the percolation theory at their respective phase transition points; these are both families of classes, one for each lattice dimension. Typically, a family of universality classes will have a lower and upper critical dimension: below the lower critical dimension, the universality class becomes degenerate, and above the upper critical dimension the critical exponents stabilize and can be calculated by an analog of mean field theory.

List of critical exponents

Critical exponents are defined in terms of the variation of certain physical properties of the system near its phase transition point. These physical properties will include its reduced temperature, its order parameter measuring how much of the system is in the "ordered" phase, the specific heat, and so on.
For symmetries, the group listed gives the symmetry of the order parameter. The group is the dihedral group, the symmetry group of the n-gon, is the n-element symmetric group, is the octahedral group, and is the orthogonal group in n dimensions. 1 is the trivial group.
dimensionSymmetriesclass
23-state Potts
2Ashkin-Teller
11011Ordinary percolation
21−Ordinary percolation
31−0.6250.41811.7935.290.876190.46 or 0.59Ordinary percolation
41−0.7560.6571.4220.689−0.0944Ordinary percolation
510.8301.1850.569Ordinary percolation
61−11120Ordinary percolation
110.1594640.2764862.2777300.1594641.0968540.313686Directed percolation
210.4510.5361.600.4510.7330.230Directed percolation
310.730.8131.250.730.5840.12Directed percolation
41−11120Directed percolation
3−0.120.3661.3950.7070.035Heisenberg
201512D Ising
30.110070.326531.23734.78930.630120.036393D Ising
Local linear interface
allany0130Mean field
Molecular beam epitaxy
Gaussian free field
3−0.01460.34851.31774.7800.671550.0380XY