Phase Transitions §

Global idea. The order parameter is the macroscopic variable that distinguishes the two phases: it measures how much symmetry the disordered phase has that the ordered phase lacks. It transforms under the symmetry group that is broken. A ferromagnet breaks $\mathbb{Z}_2$ spin-flip symmetry; a liquid–gas transition breaks the density-translation symmetry. The choice of order parameter is not unique — any quantity that vanishes in the disordered phase and is nonzero in the ordered phase works.

The Landau picture. Write the free energy as an analytic function of the order parameter, constrained by symmetry: $F(\phi) = a(T-T_c)\phi^2 + b\phi^4 + \cdots$ with $b > 0$. Above $T_c$ the coefficient of $\phi^2$ is positive, so the minimum sits at $\phi = 0$ (disordered). Below $T_c$ it is negative, so two symmetric minima appear at $\phi = \pm\sqrt{a(T_c-T)/(2b)}$ (ordered), giving the mean-field prediction $\phi \sim (T_c - T)^{1/2}$.

Example. For a ferromagnet, $\phi$ is the net magnetization. Above the Curie temperature, thermal fluctuations point the atomic spins in random directions and $M = 0$. Below it, the spin-spin coupling wins and the system spontaneously picks one of the two minima — a symmetry breaking choice that the Hamiltonian itself does not make.

The double well below $T_c$ is the whole story of spontaneous symmetry breaking: the system must choose one minimum, and an infinite system cannot tunnel between them.

Authoritative sources. Landau’s original 1937 paper (republished, English translation); Cardy, Scaling and Renormalization in Statistical Physics, ch. 4.

Global idea. Power laws are the fingerprint of scale invariance. The correlation length $\xi$ is the distance over which fluctuations of the order parameter are correlated. At the critical point, $\xi \to \infty$, and the system becomes scale-invariant: fluctuations exist on all length scales, which is exactly why a power law (the only scale-free functional form) describes the singularities. This is why critical phenomena are described by the renormalization group (RG): integrating out short-distance modes reveals the same physics at longer scales.

Example. Cool a ferromagnet toward its Curie point from above. Domains of aligned spins grow from a few lattice spacings to macroscopic size ($\xi \to \infty$); the susceptibility $\chi$ — the response to an external field — diverges because a whole correlated region flips coherently; the specific heat $C$ peaks (in 2D Ising it diverges only logarithmically, $C \sim \ln|t|$, corresponding to $\alpha = 0$).

Scaling relations. The exponents are not independent. The hyperscaling relation $\alpha + 2\beta + \gamma = d\nu$ connects them, and it holds below the upper critical dimension. Mean-field theory (Landau) gives $\alpha = 0, \beta = 1/2, \gamma = 1, \nu = 1/2$ and is exact only for $d \geq 4$ (the upper critical dimension). For the 2D Ising model, exact results (Onsager, Yang) give $\alpha = 0, \beta = 1/8, \gamma = 7/4, \nu = 1$.

Authoritative sources. Onsager, Crystal Statistics I (1944); Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, ch. 3.

Global idea. The type of transition is determined by the behavior of the free energy $F$ at the transition. If $F$ itself is continuous but a first derivative (entropy $S = -\partial F/\partial T$, volume $V = \partial F/\partial P$) jumps, the transition is first-order: latent heat $L = T_c \Delta S$ is released, and two phases coexist in equilibrium. If the first derivatives are continuous but a second derivative (specific heat, compressibility, susceptibility) diverges, the transition is second-order (continuous).

Example. In the phase diagram above, follow the boiling curve from the triple point upward: crossing it at any pressure below the critical pressure is a first-order transition (water jumps from density $\rho_g$ to $\rho_l$, absorbing latent heat). But the curve ends at the critical point — above it there is no distinction between liquid and gas, and approaching that endpoint along the curve gives a continuous transition with diverging compressibility and critical exponents. The melting line, by contrast, has no known critical point: solid and liquid differ by a symmetry (positional order), not just by degree.

Quantum and topological transitions. Quantum phase transitions occur at $T = 0$ and are controlled by a non-thermal parameter; at finite $T$ they are rounded over a scale $T \sim \xi^{-z}$ where $z$ is the dynamic critical exponent. Topological transitions (e.g., the Kosterlitz–Thouless transition in 2D superfluids) involve the unbinding of topological defects rather than symmetry breaking — the 2016 Nobel prize recognized this work.

Authoritative sources. Kosterlitz & Thouless, Ordering, metastability and phase transitions in two-dimensional systems (1973); Nobel Prize 2016: topological phase transitions.

Global idea. Coarse-graining replaces blocks of spins by a single effective spin and rescales lengths back. Repeating this generates a flow in the space of Hamiltonians. Criticality means scale invariance, and scale invariance means the flow must reach a fixed point ($\mathcal{R}[H^*] = H^*$): the system looks the same at every scale. Near a fixed point, perturbations decompose into relevant directions (they grow under coarse-graining and push the system away from criticality — the reduced temperature $t$ is one) and irrelevant directions (they shrink and die out). The rate of growth/decay is the critical exponent: $\nu = 1/y_t$ where $y_t$ is the relevant eigenvalue.

Example. An Ising ferromagnet, a liquid–gas system, and a lattice gas look nothing alike microscopically, yet all three flow to the same fixed point under coarse-graining — hence they share the exact same exponents ($\beta = 1/8$ etc. in 2D). That is universality, and it is why studying one lattice model teaches us about real fluids.

Computing exponents. The epsilon expansion ($\epsilon = 4 - d$) computes exponents perturbatively around the upper critical dimension; the Monte Carlo RG and tensor network methods provide non-perturbative results in low dimensions.

Authoritative sources. Wilson & Kogut, The Renormalization Group and the ε Expansion (1974); Cardy, Scaling and Renormalization in Statistical Physics, ch. 3.

A 2D Ising ferromagnet has critical temperature $T_c = 2.269\, J/k_B$. Estimate the correlation length at $T = 0.99\, T_c$ using $\nu = 1$ (exact for 2D Ising). Then estimate it at $T = 0.999\, T_c$ and explain what the trend means physically.