Critical Exponents, Scaling, and Landau Theory

Near a continuous transition the response functions diverge and the order parameter vanishes, and each does so as a power of the distance from the critical point. Measure the temperature from $T_c$ by the dimensionless reduced temperature

and the singular behavior is captured by a small set of critical exponents. The exponents, not the value of $T_c$, are what different systems share: the same handful of numbers describes the liquid-gas critical point of argon, the Curie point of nickel, and the order-disorder transition of a brass alloy. Landau theory computes the exponents from an expansion of the free energy, gets them uniformly wrong below four dimensions, yet organizes the whole subject, because the way it fails points to what the correct theory must supply.

The critical exponents

Six exponents describe the leading singularities of a magnet near its critical point; the fluid dictionary maps magnetization to density difference, field to pressure deviation, and susceptibility to compressibility.

• Specific heat, $\alpha$: $C\sim |t{|}^{-\alpha }$ as $t\to 0$ at zero field.
• Order parameter, $\beta$: $m\sim (-t{)}^{\beta }$ for $t<0$ at zero field.
• Susceptibility, $\gamma$: $\chi =(\partial m/\partial h{)}_{h=0}\sim |t{|}^{-\gamma }$.
• Critical isotherm, $\delta$: $m\sim h^{1/\delta }$ at $t=0$.
• Correlation length, $\nu$: $\xi \sim |t{|}^{-\nu }$.
• Correlation function at $T_c$, $\eta$: $G(r)\sim r^{-(d-2+\eta )}$ at $t=0$ in $d$ dimensions.

Each exponent isolates one direction of approach to the critical point. Two, $\nu$ and $\eta$, describe the spatial structure of fluctuations through the correlation length and the decay of the correlation function; the other four describe thermodynamic quantities. All six are pure numbers, and their values are the experimental content of critical phenomena.

The Landau expansion

Landau's method builds the free energy directly from the order parameter and the symmetry, without reference to a microscopic Hamiltonian. Take the order parameter $m$ to be small near $T_c$ and expand the free energy density in powers of it, keeping only terms allowed by the up-down symmetry $m\to -m$ of the disordered phase:

with $b>0$ so the free energy is bounded below. The odd term is present only through the explicit field coupling $-hm$. The single assumption that produces a transition is that the quadratic coefficient changes sign at $T_c$, positive above and negative below, so to leading order

The equilibrium magnetization minimizes $f$, so $\partial f/\partial m=0$ gives the equation of state

Mean-field exponents

The four thermodynamic exponents follow from the Landau equation of state by direct computation.

• Order parameter. At $h=0$ and $t<0$, the nonzero minimum solves $m^2=-a/2b=-a_{0}t/2b$, so $m=(-a_{0}t/2b{)}^{1/2}\propto (-t{)}^{1/2}$ and $\beta =1/2$.
• Susceptibility. Differentiating $2am+4b{m}^3=h$ at fixed $T$, ${\chi }^{-1}=2a+12b{m}^2$. Above $T_c$, $m=0$ gives $\chi =1/2a_{0}t\propto t^{-1}$; below $T_c$, $m^2=-a/2b$ gives $\chi =1/(-4a)\propto |t{|}^{-1}$. Either way $\gamma =1$.
• Critical isotherm. At $t=0$ the quadratic term vanishes, so $4b{m}^3=h$, $m\propto h^{1/3}$, and $\delta =3$.
• Specific heat. The minimum free energy below $T_c$ is $f_{\min }=-a^2/4b=-a_0^2{t}^2/4b$; above $T_c$ it is $f_0$. The second temperature derivative jumps by a finite amount at $T_c$, a discontinuity rather than a divergence, so $\alpha =0$.

Adding a gradient term $c(\nabla m{)}^2$ to the free energy (the Ornstein-Zernike theory) fixes the two correlation exponents: the correlation length diverges as $\xi \propto |t{|}^{-1/2}$, giving $\nu =1/2$, and the correlation function at $T_c$ decays as the bare Coulomb-like form $r^{-(d-2)}$, giving $\eta =0$. The full mean-field set is

The failure against experiment

Mean-field exponents are wrong wherever fluctuations matter, which is every real system in three dimensions or fewer. The exact two-dimensional Ising values and careful measurements on three-dimensional fluids and magnets disagree with the mean-field numbers across the board.

The three-dimensional Ising exponents match measured values on liquid-gas critical points and uniaxial ferromagnets to within experimental error, which is the quantitative statement of universality: systems with the same order-parameter symmetry and the same spatial dimension share exponents regardless of microscopic detail. Mean-field theory sits in none of these columns except at and above four dimensions, where fluctuations are weak enough that its neglect of them is harmless.

Scaling relations

The six exponents are not independent. Thermodynamic inequalities, provable from convexity, become equalities under the scaling hypothesis: near $T_c$ the singular part of the free energy is a generalized homogeneous function of $t$ and $h$, so a single rescaling of both collapses all the singular behavior. The hypothesis forces exact relations among the exponents.¹

• Rushbrooke: $\alpha +2\beta +\gamma =2$.
• Widom: $\gamma =\beta (\delta -1)$.
• Fisher: $\gamma =\nu (2-\eta )$.
• Josephson (hyperscaling): $\nu d=2-\alpha$.

Two of the six exponents fix the rest. The mean-field set satisfies the first three identically ($0+1+1=2$; $1=\frac{1}{2}\cdot 2$; $1=\frac{1}{2}\cdot 2$) but satisfies Josephson only at $d=4$: $\nu d=\frac{1}{2}d=2$ requires $d=4$. Hyperscaling involves the dimension explicitly and holds only below the upper critical dimension; above it, fluctuations decouple and mean-field exponents take over while hyperscaling breaks. This is the exponent-level statement that four dimensions is the boundary of mean-field validity.

The correlation length and universality classes

The correlation length is the single length scale that governs the critical region. Away from $T_c$ it is finite: fluctuations are correlated over a distance $\xi$ and independent beyond it. As $t\to 0$ it diverges as $\xi \sim |t{|}^{-\nu }$, and at the critical point fluctuations exist on every scale from the lattice spacing to the system size. The divergence of $\xi$ is the root cause of the other divergences — response functions integrate correlations over the whole correlated volume, which blows up with $\xi$ — and it is the physical reason microscopic details wash out: when fluctuations span all scales, the short-scale structure of the interaction cannot set the exponents.

Systems that share the same critical exponents form a universality class, fixed by two properties alone:

• Spatial dimension $d$ — the number of dimensions the fluctuations explore.
• Order-parameter symmetry — the symmetry broken at the transition, together with the number of order-parameter components (a scalar for the Ising class, a two-component vector for the XY class of superfluids, a three-component vector for the Heisenberg class of isotropic magnets).

Everything else — the lattice geometry, the range and shape of short-ranged interactions, the microscopic constituents — is irrelevant to the exponents. The uniaxial ferromagnet and the liquid-gas critical point share exponents because both have a one-component (scalar) order parameter in three dimensions: both belong to the three-dimensional Ising class. Explaining why only $d$ and the symmetry survive, and computing the exponents each class carries, is the achievement of the renormalization group.

Summary

• Near a continuous transition, singular quantities follow power laws in the reduced temperature $t=(T-T_c)/T_c$, encoded by the exponents $\alpha ,\beta ,\gamma ,\delta$ (thermodynamic) and $\nu ,\eta$ (correlations).
• Landau theory expands the free energy $f=a_{0}t{m}^2+b{m}^4-hm$ in the order parameter; minimizing it gives the mean-field exponents $\alpha =0$, $\beta =1/2$, $\gamma =1$, $\delta =3$, $\nu =1/2$, $\eta =0$.
• These disagree with the exact 2D Ising values and with 3D experiments; the 3D Ising exponents match measured fluid and magnet critical points, the content of universality.
• The scaling relations — Rushbrooke $\alpha +2\beta +\gamma =2$, Widom $\gamma =\beta (\delta -1)$, Fisher $\gamma =\nu (2-\eta )$, Josephson $\nu d=2-\alpha$ — leave only two exponents independent; hyperscaling holds only for $d<4$.
• The correlation length $\xi \sim |t{|}^{-\nu }$ diverges at $T_c$, making fluctuations scale-free and washing out microscopic detail; universality classes are fixed by spatial dimension and order-parameter symmetry alone.