Phase Transitions: Order Parameters, the Ising Model & Critical Exponents

Ehrenfest's classification: first-order vs. second-order

Paul Ehrenfest classified phase transitions in 1933 by the lowest derivative of the Gibbs free energy G(T,P) that is discontinuous at the transition. In a first-order transition — ice melting, water boiling — G itself stays continuous but its first derivatives (entropy and volume) jump, and the system absorbs or releases latent heat at a fixed temperature. In a second-order (continuous) transition — the ferromagnetic Curie point, the zero-field superconducting transition — G and its first derivatives stay continuous, but a second derivative like heat capacity diverges. There is no latent heat; the change is gradual. Water's own liquid-gas line ends at a critical point (T_c = 374.14 °C, P_c = 220.64 bar) where the first-order distinction between liquid and gas vanishes entirely.

The order parameter and spontaneous symmetry breaking

Lev Landau introduced the order parameter ψ in 1937 to quantify how ordered a phase is: zero in the disordered high-temperature phase, non-zero in the ordered low-temperature phase, continuous at second-order transitions but discontinuous at first-order ones. A ferromagnet's Hamiltonian is symmetric under flipping every spin, yet below T_c the system spontaneously picks one direction of magnetisation, breaking that symmetry — a phenomenon called spontaneous symmetry breaking. When the broken symmetry is continuous (like the phase of a superconducting wavefunction), Goldstone's theorem guarantees massless excitations — spin waves in magnets, phonons in superfluids.

The 2D Ising model and Onsager's exact solution

The Ising model is the simplest system that shows a genuine phase transition: binary spins σᵢ ∈ {−1,+1} on a lattice with Hamiltonian H = −J Σσᵢσⱼ − h Σσᵢ. The 1D chain (solved by Ising himself in 1925) has no transition at any T > 0. The 2D square lattice is different — Lars Onsager's landmark 1944 exact solution gives a genuine critical temperature:

k_B T_c / J = 2 / ln(1 + √2) ≈ 2.2692
M(T)  ~ (1 − T/T_c)^β,  β = 1/8   (spontaneous magnetisation)
C(T)  ~ −ln|1 − T/T_c|             (logarithmic heat-capacity divergence)

This is one of the only exactly solved interacting statistical-mechanics models in two dimensions and remains a landmark of 20th-century theoretical physics. Below T_c the lattice spontaneously magnetises into large aligned domains; above it, thermal noise scrambles the spins into a disordered paramagnet.

Critical exponents and universality

Near a second-order critical point, thermodynamic quantities diverge or vanish as power laws in the reduced temperature t = (T−T_c)/T_c: magnetisation M ~ |t|^β, susceptibility χ ~ |t|^(−γ), correlation length ξ ~ |t|^(−ν). For the 2D Ising model these exponents are β = 1/8, γ = 7/4, ν = 1, α = 0 — and remarkably, the same exponents describe every system with the same spatial dimension and order-parameter symmetry, regardless of microscopic detail. That is universality: iron and nickel share identical critical exponents as 3D Ising-class magnets despite very different electronic structure, and the liquid-gas critical point vanishes with 3D Ising exponents (β ≈ 0.326) even though it has nothing to do with magnetism. At the critical point the correlation length diverges and the system becomes scale-invariant at every zoom level — a phenomenon visible as scale-free domain clusters, called critical opalescence.

Frequently asked questions

Try it live

Everything above runs in your browser — open Phase Transition — Order Parameter & Critical Point and drag the temperature slider slowly across Tc ≈ 2.269 to watch order collapse into disorder, live, on a Metropolis Monte Carlo Ising lattice.