Universality Classes
At a continuous phase transition, thermodynamic quantities diverge as power laws characterised by critical exponents. A universality class is the set of all systems that share the same exponents — independent of microscopic details.
Membership depends only on: (1) spatial dimension d, (2) symmetry of the order parameter, (3) range of interactions.
Class index
| Class | Description | Models | Page |
|---|---|---|---|
Ising | Z₂ symmetry, d=2,3,≥4 | IsingSquare, IsingTriangular, RandomBondIsing2D, … | → |
Percolation | Geometric transition, d=2,3,≥6 | — | → |
Potts | S₃/S₄ symmetry, d=2 exact | — | → |
KPZ | Non-equilibrium growth, 1+1D | — | → |
XY | O(2) symmetry, d=2,3,≥4 | — | → |
Heisenberg | O(3) symmetry, d=3,≥4 | — | → |
MeanField | Baseline, d ≥ d_c | CurieWeissIsing | → |
E8 | Exact mass ratios, integrable | — | → |
CardyEntanglement | Entanglement scaling at CFT QCPs | — | → |
MinimalModel | M(p,p') rational CFT, c < 1 | TricriticalIsing, TricriticalPotts3, YangLee | → |
WZW | SU(2)_k WZW, Sugawara c | ChernSimons3D | → |
RMT | Wigner-Dyson level statistics, β=1,2,4 | — | → |
Poisson | Integrable/MBL baseline | — | → |
The Universality{C} API
using QAtlas
# 2D Ising critical exponents (exact Rationals)
e = QAtlas.fetch(Universality(:Ising), CriticalExponents(); d=2)
# => (β = 1//8, ν = 1//1, γ = 7//4, η = 1//4, δ = 15//1, α = 0//1, c = 1//2)
# Mean-field baseline
e_mf = QAtlas.fetch(MeanField(), CriticalExponents())
# Rational CFT: Ising minimal model M(3,4)
c = QAtlas.fetch(MinimalModel(3,4), CentralCharge())Scaling relations
For any class with standard exponents:
\[\alpha + 2\beta + \gamma = 2 \qquad \text{(Rushbrooke)}\]
\[\gamma = \beta(\delta - 1) \qquad \text{(Widom)}\]
\[\gamma = \nu(2 - \eta) \qquad \text{(Fisher)}\]
\[2 - \alpha = d\nu \qquad \text{(Josephson hyperscaling, } d < d_c\text{)}\]
QAtlas stores exact-class exponents as Rational{Int}, so these identities can be verified with zero floating-point error in the test suite.