Universality Classes
Overview
At a continuous phase transition, thermodynamic quantities diverge (or vanish) as power laws characterised by critical exponents. A universality class is the set of all systems that share the same critical exponents. Membership depends only on:
- Spatial dimension $d$
- Symmetry of the order parameter
- Range of interactions (short-range vs long-range)
Microscopic details –- lattice structure, coupling strength, even the distinction between classical and quantum –- are irrelevant. This remarkable fact, grounded in the renormalization group, is the reason a single library entry can serve as a reference for an entire family of physical models.
Universality Classes in QAtlas
| Class | Symbol | Dimensions | Type | Key feature | Page |
|---|---|---|---|---|---|
| Ising | :Ising | $d = 2, 3, \geq 4$ | Exact / Bootstrap / MF | $\mathbb{Z}_2$ symmetry | → |
| Percolation | :Percolation | $d = 2, 3, \geq 6$ | Exact / MC / MF | Geometric transition | → |
| 3-state Potts | :Potts3 | $d = 2$ | Exact | $S_3$ symmetry | → |
| 4-state Potts | :Potts4 | $d = 2$ | Exact | Marginal, log corrections | → |
| KPZ | :KPZ | $1+1$D | Exact | Non-equilibrium growth | → |
| XY | :XY | $d = 2, 3, \geq 4$ | BKT / Bootstrap / MF | $\mathrm{O}(2)$ symmetry | → |
| Heisenberg | :Heisenberg | $d = 3, \geq 4$ | Bootstrap / MF | $\mathrm{O}(3)$ symmetry | → |
| Mean-Field | MeanField() | $d \geq d_c$ | Exact | Baseline reference | → |
| E8 | :E8 | — | Exact mass ratios | Integrable field theory | → |
The Universality{C} API
All universality data in QAtlas is accessed through the type-safe Universality{C} interface:
using QAtlas
# Construct a universality-class handle
u = Universality(:Ising) # equivalent to Universality{:Ising}()
# Fetch critical exponents for a given dimension
e = QAtlas.fetch(u, 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 has its own type (no dimension needed)
e_mf = QAtlas.fetch(MeanField(), CriticalExponents())Return types depend on whether the exponents are analytically known:
| Situation | Return type | Example |
|---|---|---|
| Exact ($d = 2$ Ising, Potts, ...) | Rational{Int} | β = 1//8 |
| Numerical ($d = 3$ bootstrap, MC) | Float64 + _err fields | β = 0.32642, β_err = 1e-5 |
| Mean-field ($d \geq d_c$) | Rational{Int} | β = 1//2 |
Scaling Relations
For any universality class with standard critical exponents, the following relations hold (see calculation note):
\[\alpha + 2\beta + \gamma = 2 \qquad \text{(Rushbrooke)}\]
\[\gamma = \beta(\delta - 1) \qquad \text{(Widom)}\]
\[\gamma = \nu(2 - \eta) \qquad \text{(Fisher)}\]
\[2 - \alpha = d\nu \qquad \text{(Josephson hyperscaling, } d < d_c\text{)}\]
QAtlas stores exact-class exponents as Rational{Int}, so these identities can be verified with zero floating-point error in the test suite.
Further Reading
- Ising –- the most thoroughly studied class
- Mean-Field –- baseline for $d \geq d_c$
- Cross-Verification –- how universality predictions are tested against model-specific results