Critical Exponents

Definition

Near a continuous phase transition — the Curie point of a magnet, the critical point of a fluid, the lambda transition of helium — thermodynamic quantities do not jump. They diverge or vanish as power laws. The exponents of those power laws are the critical exponents.

The natural variable is the reduced temperature, the fractional distance from the critical temperature T_c:

t = (T − Tc) / Tc

In terms of t, the four headline quantities behave like:

Order parameter   M  ~ (−t)^β       (t < 0, ordered side)
Susceptibility    χ  ~ |t|^(−γ)     (diverges at Tc)
Correlation len.  ξ  ~ |t|^(−ν)     (diverges at Tc)
Heat capacity     C  ~ |t|^(−α)     (cusp or weak divergence)

The order parameter M is whatever distinguishes the two phases — net magnetization for a ferromagnet, density difference for a fluid, sublattice imbalance for an alloy. Above T_c it is zero (disordered); below T_c it grows from zero like (−t)^β. The susceptibility measures how strongly the system responds to a small field, and it blows up exactly at T_c. The correlation length ξ is the typical size of a correlated fluctuation — a patch of aligned spins — and it too diverges, which is the deep reason for everything else.

How it works: the diverging correlation length

Far from T_c a magnet is either uniformly ordered or a fizz of tiny uncorrelated domains a few atoms across. As T → Tc, fluctuating ordered patches grow without bound: ξ → ∞. At the critical point fluctuations exist on every length scale at once. A snapshot of the spins looks statistically the same whether you zoom in or out — the system is scale invariant.

Scale invariance is the engine of universality. If the only special length, ξ, has diverged, then microscopic lengths (the lattice spacing, the range of the interaction) no longer matter. Coarse-graining the system — averaging spins in blocks and rescaling — leaves its long-distance physics unchanged. This is Kadanoff's block-spin picture, made rigorous by Wilson's renormalization group (RG), which earned the 1982 Nobel Prize. The fixed point of the RG flow controls the transition, and its handful of relevant eigenvalues fix the exponents. Everything irrelevant — lattice type, exact coupling — flows away.

The practical consequence is stunning: a few hundred grams of nickel near 627 K and a sealed tube of CO₂ near 31 °C and 73 atm obey the same power laws with the same numbers, because both sit in the 3D Ising universality class (scalar order parameter, d = 3).

A worked example: reading β off the data

Suppose we measure the magnetization of an iron-like sample at temperatures below its Curie point T_c = 1000 K, in units where the saturation magnetization is 1. We expect M = M₀ · (−t)^β with t = (T − 1000)/1000.

T (K)	−t = (Tc−T)/Tc	Measured M	M from β=0.326, M₀=1.3
999	0.001	0.130	0.130
990	0.010	0.275	0.275
950	0.050	0.488	0.488
900	0.100	0.613	0.613
800	0.200	0.768	0.768
500	0.500	1.038	1.038

Take logs. From the first two rows, −t grows by a factor of 10 (0.001 → 0.010) while M grows by a factor 0.275/0.130 = 2.12. The slope on a log-log plot is log(2.12) / log(10) = 0.326. That slope is β. Notice the mean-field prediction β = 1/2 would have made M grow by a factor √10 ≈ 3.16 over the same decade — clearly steeper than the data. The smaller real exponent means fluctuations let order build up faster than mean-field predicts as you cool below T_c.

Repeat the trick for susceptibility above T_c: a decade closer to T_c multiplies χ by 10^γ = 10^1.237 ≈ 17.3. For the correlation length the factor is 10^ν = 10^0.630 ≈ 4.27 per decade. Those slopes are exactly what the visualization above fits live as you drag the temperature.

Variants and regimes: the major universality classes

Universality classes are labeled by the spatial dimension d and the number of order-parameter components n (the symmetry). A few classes cover an enormous range of real systems.

Universality class	β	γ	ν	α	Physical examples
Mean-field (d ≥ 4)	0.500	1.000	0.500	0.000	Superconductors, d≥4 models, Landau theory
2D Ising (exact, Onsager)	0.125	1.750	1.000	0 (log)	Thin magnetic films, adsorbed monolayers
3D Ising (n=1)	0.326	1.237	0.630	0.110	Water/CO₂ critical point, uniaxial magnets, binary alloys
3D XY (n=2)	0.349	1.317	0.672	−0.015	Superfluid helium-4 lambda point
3D Heisenberg (n=3)	0.366	1.396	0.711	−0.134	Isotropic ferromagnets (EuO, Fe, Ni)
3D percolation	0.418	1.793	0.876	—	Connectivity transitions, gelation, porous media

Two patterns jump out. First, raising the dimension toward 4 pushes the exponents toward the mean-field values — fluctuations matter less in higher dimensions because each spin has more neighbors to average over. Second, within a dimension the exponents drift up as n grows (Ising → XY → Heisenberg), because a continuous order-parameter symmetry has more directions for fluctuations to explore. The negative α in the XY and Heisenberg classes means the heat capacity has a finite cusp rather than a true divergence.

Scaling relations and hyperscaling

The exponents are not free parameters. Scaling theory derives them from just two independent numbers, leaving four exact relations:

Rushbrooke:  α + 2β + γ = 2
Widom:       γ = β(δ − 1)
Fisher:      γ = ν(2 − η)
Hyperscaling: 2 − α = d·ν      (involves the dimension d)

Plug in the 3D Ising set as a consistency check:

Rushbrooke:  0.110 + 2(0.326) + 1.237 = 0.110 + 0.652 + 1.237 = 1.999  ≈ 2 ✓
Hyperscaling: 2 − 0.110 = 1.890   and   3 × 0.630 = 1.890            ✓

The agreement to three decimals is not luck — it is forced by the RG structure. If you ever measure a set of exponents that violates these relations badly, you have either mis-located T_c or you are not in the asymptotic critical region. Note that hyperscaling fails above d = 4, exactly where mean-field takes over: there 2 − α = 2 but d·ν = 4 × 0.5 = 2 only at d = 4, and for d > 4 the dimension drops out and the exponents stick at their mean-field values.

JavaScript — fit a critical exponent from data

// Fit an exponent by linear regression on log-log axes.
// data: array of { t, y } with t = |reduced temperature|, y = quantity.
// Returns the slope (the exponent, up to sign) and the amplitude.
function fitExponent(data) {
  const pts = data
    .filter(d => d.t > 0 && d.y > 0)
    .map(d => ({ x: Math.log(d.t), y: Math.log(d.y) }));
  const n = pts.length;
  const sx  = pts.reduce((s, p) => s + p.x, 0);
  const sy  = pts.reduce((s, p) => s + p.y, 0);
  const sxx = pts.reduce((s, p) => s + p.x * p.x, 0);
  const sxy = pts.reduce((s, p) => s + p.x * p.y, 0);
  const slope = (n * sxy - sx * sy) / (n * sxx - sx * sx);
  const intercept = (sy - slope * sx) / n;
  return { exponent: slope, amplitude: Math.exp(intercept) };
}

// Synthetic magnetization below Tc with the true 3D Ising beta.
const beta = 0.326, M0 = 1.3;
const mag = [0.001, 0.01, 0.05, 0.1, 0.2, 0.5]
  .map(t => ({ t, y: M0 * Math.pow(t, beta) }));

console.log(fitExponent(mag));
// { exponent: 0.326, amplitude: 1.3 }  -> recovers beta exactly

// Susceptibility above Tc diverges, so y grows as t shrinks: exponent is negative.
const gamma = 1.237;
const chi = [0.2, 0.1, 0.05, 0.01, 0.005, 0.001]
  .map(t => ({ t, y: 0.8 * Math.pow(t, -gamma) }));
console.log(fitExponent(chi).exponent); // -1.237  -> magnitude is gamma

// Verify the hyperscaling relation 2 - alpha = d * nu in 3D.
const alpha = 0.110, nu = 0.630, d = 3;
console.log((2 - alpha).toFixed(3), (d * nu).toFixed(3)); // 1.890 1.890

The whole measurement reduces to a straight-line fit — but only after T_c is pinned down, because t sits inside the logarithm. Get T_c wrong and the log-log plot bends, biasing the slope.

Where critical exponents show up

• Fluids at the critical point. Water at 374 °C / 218 atm and CO₂ at 31 °C / 73 atm show critical opalescence — light scatters off density fluctuations as ξ diverges — and follow the 3D Ising exponents.
• Ferromagnets near the Curie point. Iron (T_c = 1043 K), nickel (627 K), and gadolinium lose magnetization as (−t)^β; the universality class depends on the spin anisotropy.
• Superfluid helium-4. The lambda transition at 2.17 K is the cleanest 3D XY experiment ever done — a microgravity Space Shuttle experiment measured ν to better than 0.1%.
• Binary alloys and liquid mixtures. Order-disorder transitions in brass and demixing of oil-water-like mixtures share the Ising exponents.
• Percolation and networks. The onset of a spanning cluster — gelation, forest fires, epidemic thresholds, conductivity of composites — has its own exponents.
• Beyond equilibrium. The same scaling machinery appears in directed percolation, the depinning of interfaces, and even some models of neural avalanches and financial crashes.

Common pitfalls and misconceptions

• Confusing first-order with continuous transitions. Ice melting at 0 °C is first-order: a latent heat, a jump in density, a finite correlation length, and no critical exponents. Exponents live only at continuous transitions and at the critical endpoint of a first-order line.
• Trusting data outside the critical region. Power laws are asymptotic. Corrections-to-scaling terms like (1 + b·|t|^θ) matter for |t| ≳ 0.01, so fitting far from T_c gives effective exponents that drift toward mean-field.
• Getting T_c wrong. Because the exponent is the slope of log y versus log|t|, a biased T_c curves the line and corrupts the slope. Always fit T_c, amplitude, and exponent together.
• Expecting mean-field to work in 3D. Landau theory is seductive and gives clean fractions (β = 1/2), but fluctuations dominate below the upper critical dimension d = 4. In real 3D systems β ≈ 0.326, not 0.5.
• Thinking the amplitudes are universal. The exponents are universal; the prefactors (amplitudes) are not — though certain amplitude ratios are. Two systems in one class share β but not the raw value of M₀.
• Assuming every divergence is a phase transition. A peak in susceptibility on a finite system or simulation is rounded and shifted; only in the thermodynamic limit does a true power-law singularity appear. Finite-size scaling is the tool that extracts exponents from finite samples.

Derivation analysis: Landau theory vs the real world

Landau theory writes the free energy as a power series in the order parameter, keeping only the terms allowed by symmetry:

F(M) = F₀ + a·t·M² + b·M⁴ + ...        (b > 0)

Minimizing, ∂F/∂M = 2a·t·M + 4b·M³ = 0. For t < 0 the nonzero solution is M² = −a·t/(2b), so M ~ (−t)^(1/2) and β = 1/2. Differentiating again gives the susceptibility χ ~ |t|^(−1), so γ = 1, and the same machinery yields ν = 1/2, α = 0, δ = 3. These are the mean-field exponents.

So why are the measured 3D numbers different (β = 0.326 instead of 0.500)? Because Landau theory assumed M is uniform — it ignored fluctuations. The Ginzburg criterion estimates when that assumption breaks: fluctuations dominate whenever the system is below the upper critical dimension d = 4. Below d = 4, the correlation length's divergence means fluctuations on all scales contribute, and the RG replaces the naive Landau exponents with non-trivial ones. The ε-expansion (ε = 4 − d) treats the difference perturbatively: to first order, ν ≈ 1/2 + ε/12, which for ε = 1 gives ν ≈ 0.583, already moving in the right direction toward the true 0.630. Higher-order resummation and the modern conformal bootstrap nail the values to many digits.

Exponent	Mean-field	3D Ising (true)	What it controls
β	0.500	0.326	How order parameter grows below Tc
γ	1.000	1.237	How susceptibility diverges at Tc
ν	0.500	0.630	How correlation length diverges
α	0.000	0.110	Heat-capacity singularity
δ	3.000	4.790	M vs field at exactly Tc
η	0.000	0.036	Decay of correlations at Tc

The takeaway: mean-field theory gets the qualitative picture right — a continuous transition with power laws — and the exact exponents only at d ≥ 4. The gap between 0.500 and 0.326 is the fingerprint of fluctuations, and explaining it precisely is one of the great triumphs of twentieth-century theoretical physics.