Mean-Field (Landau)
Overview
The Landau mean-field theory provides the baseline reference for critical exponents. It describes the behaviour of any system above its upper critical dimension $d_c$, where fluctuations are irrelevant and the saddle-point approximation to the partition function becomes exact.
The mean-field exponents are independent of spatial dimension (for $d \geq d_c$), symmetry group, and microscopic details. They represent the simplest possible universality class and serve as the starting point for $\epsilon$-expansion calculations ($d = d_c - \epsilon$).
Critical Exponents
| Exponent | Value | Physical meaning |
|---|---|---|
| $\alpha$ | $0$ | Specific heat: finite jump (discontinuity), no divergence |
| $\beta$ | $1/2$ | Order parameter: $M \sim (T_c - T)^{1/2}$ |
| $\gamma$ | $1$ | Susceptibility: $\chi \sim \lvert T - T_c\rvert^{-1}$ |
| $\delta$ | $3$ | Critical isotherm: $M \sim h^{1/3}$ |
| $\nu$ | $1/2$ | Correlation length: $\xi \sim \lvert T - T_c\rvert^{-1/2}$ |
| $\eta$ | $0$ | No anomalous dimension: $G(r) \sim r^{-(d-2)}$ (Ornstein-Zernike) |
Derivation from Landau Free Energy
The Landau free energy functional is
\[F[m] = \int d^d x \left[\frac{r}{2}m^2 + \frac{u}{4}m^4 + \frac{c}{2}(\nabla m)^2 - hm\right]\]
where $r \propto (T - T_c)$ changes sign at the transition. Minimising $F$ with respect to $m$ (saddle-point):
\[h = 0\]
, $r < 0$: $m = \pm\sqrt{-r/u} \sim (T_c - T)^{1/2}$ gives $\beta = 1/2$.\[r = 0\]
: $um^3 = h$ gives $\delta = 3$.\[h = 0\]
: $\chi = \partial m/\partial h = 1/|r| \sim |T - T_c|^{-1}$ gives $\gamma = 1$.- The Gaussian propagator $G(k) = 1/(r + ck^2)$ gives $\xi^2 = c/|r|$, so $\nu = 1/2$ and $\eta = 0$.
Scaling Relations
All four standard scaling relations are satisfied exactly:
\[\alpha + 2\beta + \gamma = 0 + 1 + 1 = 2 \quad\checkmark\]
\[\gamma = \beta(\delta - 1) = \tfrac{1}{2}\cdot 2 = 1 \quad\checkmark\]
\[\gamma = \nu(2 - \eta) = \tfrac{1}{2}\cdot 2 = 1 \quad\checkmark\]
The Josephson (hyperscaling) relation $2 - \alpha = d\nu$ gives $d = 4$. This relation is violated for $d > 4$ because the Gaussian fixed point acquires dangerous irrelevant variables. The mean-field exponents remain valid for $d > 4$, but hyperscaling does not hold.
Upper Critical Dimensions
| Model | Symmetry | $d_c$ | Reference |
|---|---|---|---|
| Ising | $\mathbb{Z}_2$ | 4 | Ginzburg criterion |
| XY / Heisenberg | O($n$), $n \geq 2$ | 4 | Wilson-Fisher (1972) |
| Percolation | $S_q$, $q \to 1$ | 6 | Toulouse (1974) |
| Directed percolation | — | $4 + 1$ | Janssen (1981) |
QAtlas API
using QAtlas
# Mean-field exponents (no dimension needed)
e = QAtlas.fetch(MeanField(), CriticalExponents())
# (β = 1//2, ν = 1//2, γ = 1//1, η = 0//1, δ = 3//1, α = 0//1)
# Equivalently, any universality class at d ≥ d_c returns these
e_ising4 = QAtlas.fetch(Universality(:Ising), CriticalExponents(); d=4)
# → same values as MeanField()All mean-field exponents are stored as Rational{Int}, enabling exact algebraic verification of scaling relations.
References
- L. D. Landau, "On the theory of phase transitions", Zh. Eksp. Teor. Fiz. 7, 19 (1937) –- original Landau theory.
- V. L. Ginzburg, "Some remarks on phase transitions of the second kind and the microscopic theory of ferroelectric materials", Sov. Phys. Solid State 2, 1824 (1960) –- Ginzburg criterion for the validity of mean-field theory.
- K. G. Wilson, M. E. Fisher, Phys. Rev. Lett. 28, 240 (1972) –- $\epsilon$-expansion showing mean-field is exact for $d \geq 4$.
Connections
- Every universality class in QAtlas reduces to mean-field at or above its upper critical dimension.
- The $\epsilon$-expansion computes corrections to these exponents perturbatively in $\epsilon = d_c - d$.
- Scaling relations: verified in calc/ising-scaling-relations.md.