Universality class

Builds on Critical phenomena Critical exponent

Definition

Consider the critical points of lattice models of statistical mechanics. Call two critical points equivalent if:

their critical exponents $\alpha, \beta, \gamma, \delta, \nu, \eta$ are equal;
their scaling functions agree after a rescaling of the temperature, the field and the lengths by constant factors.

A universality class is an equivalence class of critical points for this relation.

Remarks

Example: the nearest-neighbour Ising model on the square lattice and on the triangular lattice. Both are exactly solved, and both have $\alpha = 0$ , $\beta = 1/8$ , $\gamma = 7/4$ , $\delta = 15$ , $\nu = 1$ , $\eta = 1/4$ ; they lie in the two-dimensional Ising class.

Non-example: the three-state Potts model on the square lattice has $\beta = 1/9$ and $\nu = 5/6$ . So clause 1 fails, and it is not in the Ising class, although both models are two-dimensional with short-range interactions. The difference comes from the symmetry group of the order parameter, $\Z/3$ or $S_3$ instead of $\Z/2$ .

Many authors use clause 1 alone, since exponents are easier to compute or measure than scaling functions.

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Tag	`JNZ8FN7B4B`
URI	`m2://JNZ8FN7B4B`
Kind	definition
Writing	W0 stub
MSC	82B27, 82B28
Defines	Universality class
Uses	Critical phenomena, Critical exponent
Changed	3 November 2026

Universality class

Definition

Remarks

Reading path

Further reading

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