Critical Phenomena and Critical Exponents

Core Idea

Near a critical point, physical quantities diverge as power laws: correlation length ξ ∝ |T−T_c|^{−ν}, order parameter m ∝ |T−T_c|^β, susceptibility χ ∝ |T−T_c|^{−γ}. Remarkably, these critical exponents are universal—they depend only on dimension and symmetry, not microscopic details. This universality is explained by the renormalization group.

Explainer

From your study of phase transitions, you know that a second-order (continuous) transition is characterized by the continuous vanishing of an order parameter — for a ferromagnet, the spontaneous magnetization m that is nonzero below T_c and zero above it. Near T_c, this vanishing is not abrupt but follows a specific functional form. The central discovery of critical phenomena is that this form is a power law: m ∝ |T − T_c|^β, where β is a dimensionless number called a critical exponent. The striking fact is not merely that power laws appear, but that β takes the same value for systems as physically different as a ferromagnet and a liquid-gas transition near its critical point — despite having completely different microscopic Hamiltonians.

Each observable quantity near T_c has its own critical exponent. The correlation length ξ measures how far apart two spins (or density fluctuations) remain correlated; it diverges as ξ ∝ |T − T_c|^{−ν}. As T → T_c from either side, correlated regions grow without bound — the system develops fluctuations on all length scales simultaneously, which is why it looks the same under a microscope and under a telescope (scale invariance). The magnetic susceptibility χ = ∂m/∂h (how much the order parameter responds to a small external field) also diverges: χ ∝ |T − T_c|^{−γ}. This divergence reflects the fact that near T_c the system is poised between ordered and disordered phases, so it responds infinitely sensitively to any perturbation. The specific heat diverges as C ∝ |T − T_c|^{−α}. These four exponents β, ν, γ, α are not independent — they obey scaling relations like the Rushbrooke identity α + 2β + γ = 2, so only two are truly free.

Universality is the profound result that all systems with the same spatial dimension d and the same symmetry of the order parameter share identical critical exponents, regardless of their microscopic details. The 3D Ising universality class (discrete up/down symmetry, three dimensions) includes both uniaxial ferromagnets and the liquid-gas critical point — β ≈ 0.326 for both, measured to three decimal places. The 3D XY class (complex order parameter, like a superfluid) has a different β ≈ 0.346. This is extraordinary: the atomic structure of helium versus a magnetic material is entirely different, yet their critical fluctuations are mathematically identical. Universality means that T_c depends on microscopic details but the exponents do not — they are determined purely by dimension and symmetry.

The key intuition for why universality holds is that near T_c the diverging correlation length ξ → ∞ means microscopic details are irrelevant. When correlated patches span millions of atoms, the behavior is governed by long-wavelength, low-frequency fluctuations, not by the specific interactions at the atomic scale. The renormalization group formalizes this by showing that under successive coarse-graining (averaging over shorter-length-scale degrees of freedom), all systems with the same symmetry flow toward the same fixed point in the space of Hamiltonians. The critical exponents are determined by the linearized flow near this fixed point — they are properties of the fixed point, not of the microscopic starting Hamiltonian. This is why two systems as different as water and iron can have the same β: they flow to the same fixed point under coarse-graining.