Ising Universality Class
Overview
The Ising universality class describes second-order phase transitions with $\mathbb{Z}_2$ symmetry breaking. It is the most fundamental universality class in statistical physics, exactly solved in $d = 2$ and determined to extraordinary precision in $d = 3$ via the conformal bootstrap.
Symmetry: $\mathbb{Z}_2$ (spin-flip $\sigma \to -\sigma$).
Models in this class: 2D classical Ising, 1+1D TFIM at $h = J$, liquid-gas critical point, binary alloy order-disorder, uniaxial ferromagnets.
CFT ($d = 2$): Virasoro minimal model $\mathcal{M}(3,4)$, central charge $c = 1/2$.
$d = 2$ — Exact Critical Exponents
All critical exponents of the 2D Ising universality class are known exactly as rational numbers.
| Exponent | Value | Physical meaning | Derivation | Reference |
|---|---|---|---|---|
| $\alpha$ | $0$ | Specific heat: $C \sim \ln\lvert T - T_c\rvert$ | Onsager exact solution | Onsager (1944) Phys. Rev. 65, 117 |
| $\beta$ | $1/8$ | Order parameter: $M \sim (T_c - T)^\beta$ | Yang magnetization formula | Yang (1952) Phys. Rev. 85, 808 |
| $\gamma$ | $7/4$ | Susceptibility: $\chi \sim \lvert T - T_c\rvert^{-\gamma}$ | From scaling or exact calculation | Fisher (1964) J. Math. Phys. 5, 944 |
| $\nu$ | $1$ | Correlation length: $\xi \sim \lvert T - T_c\rvert^{-\nu}$ | Coulomb gas mapping | den Nijs (1979) J. Phys. A 12, 1857 |
| $\eta$ | $1/4$ | Anomalous dimension: $G(r) \sim r^{-(d-2+\eta)}$ | Two-point function at $T_c$ | Kadanoff (1966) Physics 2, 263 |
| $\delta$ | $15$ | Critical isotherm: $M \sim h^{1/\delta}$ at $T_c$ | Scaling relation $\delta = 1 + \gamma/\beta$ | — |
| $c$ | $1/2$ | Central charge (CFT) | Minimal model $\mathcal{M}(3,4)$ | BPZ (1984) Nucl. Phys. B 241, 333 |
Scaling Relations (Algebraically Exact)
QAtlas stores these values as Rational{Int}, so the following scaling relations hold with zero floating-point error:
\[\alpha + 2\beta + \gamma = 2 \qquad \text{(Rushbrooke)}\]
\[\gamma = \beta(\delta - 1) \qquad \text{(Widom)}\]
\[\gamma = \nu(2 - \eta) \qquad \text{(Fisher)}\]
\[2 - \alpha = d\nu \qquad \text{(Josephson, } d = 2\text{)}\]
CFT Primary Operators
| Operator | Conformal dimension $h$ | Physical correspondence |
|---|---|---|
| $\mathbb{1}$ (identity) | $0$ | — |
| $\sigma$ (spin field) | $1/16$ | Continuum limit of $\sigma^z$ |
| $\varepsilon$ (energy density) | $1/2$ | Continuum limit of $\sigma^z\sigma^z$ |
The scaling dimensions are $\Delta = h + \bar{h} = 2h$ (diagonal in the unitary minimal model). The exponents are related to the scaling dimensions via $\eta = 2\Delta_\sigma - (d - 2) = 2 \times 1/8 = 1/4$ and $\nu = 1/(d - \Delta_\varepsilon) = 1/(2 - 1) = 1$.
$d = 3$ — Conformal Bootstrap
In $d = 3$, the exponents are not known exactly but have been determined to extraordinary precision via the conformal bootstrap program.
| Exponent | Value | Uncertainty | Reference |
|---|---|---|---|
| $\alpha$ | $0.11009$ | $\pm 0.00001$ | Kos, Poland, Simmons-Duffin, Vichi (2016) |
| $\beta$ | $0.32642$ | $\pm 0.00001$ | JHEP 08, 036, Table 2 |
| $\gamma$ | $1.23708$ | $\pm 0.00001$ | " |
| $\delta$ | $4.78984$ | $\pm 0.00001$ | " |
| $\nu$ | $0.62997$ | $\pm 0.00001$ | " |
| $\eta$ | $0.03630$ | $\pm 0.00005$ | " |
These values satisfy the scaling relations approximately (within the stated uncertainties).
$d \geq 4$ — Mean-Field
Above the upper critical dimension $d_c = 4$, fluctuations are irrelevant and the exponents take the mean-field values: $\beta = 1/2$, $\nu = 1/2$, $\gamma = 1$, $\eta = 0$, $\delta = 3$.
Cross-Verification in QAtlas
The 2D Ising exponents are not just stored values — they are cross-checked against independent QAtlas results:
| Exponent | Extracted from | Method | Test file |
|---|---|---|---|
| $\beta = 1/8$ | Yang $M(T)$ near $T_c$ | log-log slope | test_universality_cross_check.jl |
| $\nu z = 1$ | TFIM gap $\Delta(N)$ at $h = J$ | log-log regression | test_universality_cross_check.jl |
| $c = 1/2$ | TFIM entanglement $S(l)$ | Calabrese-Cardy OBC | test_entanglement_central_charge.jl |
| $\alpha = 0$ | IsingSquare specific heat near $T_c$ | ForwardDiff on $Z$ | test_universality_cross_check.jl |
Each row connects a universality-level claim (Source A: CFT) to a model-level computation (Source B: Onsager / Yang / BdG), providing independent physical validation.
QAtlas API
# d = 2: exact (Rational{Int})
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)
# d = 3: numerical (Float64 + _err)
e3 = QAtlas.fetch(Universality(:Ising), CriticalExponents(); d=3)
# (α = 0.11009, α_err = 1.0e-5, β = 0.32642, β_err = 1.0e-5, ...)
# d ≥ 4: mean-field
e4 = QAtlas.fetch(Universality(:Ising), CriticalExponents(); d=4)
# → same as fetch(MeanField(), CriticalExponents())
# Backward-compatible alias
e_old = QAtlas.fetch(Ising2D(), CriticalExponents()) # same as d=2Connections
- Models: IsingSquare, TFIM
- E8 extension: E8 mass spectrum — longitudinal field at $h = J$ produces the $E_8$ integrable field theory with 8 stable particles (Zamolodchikov, 1989).
- Potts generalization: Potts — the Ising model is the $q = 2$ case of the $q$-state Potts model.