Universality class

Definition (universality class)

Two families of models (lattice or continuum) are in the same universality class if, after appropriate identification of macroscopic observables and rescalings, they share the same asymptotic long-distance behavior near criticality. Concretely, this is reflected in agreement of:

• critical exponents (e.g., $\alpha,\beta,\gamma,\nu,\eta,\delta$),
• scaling forms (equation-of-state scaling functions, finite-size scaling functions),
• operator content/relevant perturbations around the controlling fixed point.

RG characterization

In RG terms, a universality class is associated with an (infrared) stable fixed point $g^\star$ and its spectrum of relevant directions.

• Different microscopic Hamiltonians flow under RG to the same $g^\star$.
• Microscopic differences correspond to irrelevant operators (they do not affect the scaling limit).
• Relevant perturbations correspond to the few parameters that must be tuned to reach criticality (typically temperature-like and field-like).

What data typically determine the class

While details depend on the setting, the following features often control the universality class:

• spatial dimension $d$,
• symmetry of the order parameter (e.g., $\mathbb{Z}_2$, $O(N)$),
• range of interactions (short-range vs long-range),
• nature of the order parameter and constraints (scalar vs vector, conserved vs nonconserved dynamics in dynamical settings).

Example contrasts (informal)

• The 2D Ising model and many other $\mathbb{Z}_2$-symmetric short-range models share the same critical exponents and scaling functions (same class).
• Systems with continuous symmetry in 2D may avoid conventional symmetry-breaking transitions and instead exhibit topological transitions (different class), as in the Kosterlitz–Thouless transition .

Prerequisites / cross-links

• RG fixed point
• renormalization-group (RG) transformation
• critical exponents
• scaling relations among exponents
• Ising model
• continuous symmetries on spin systems