---
title: Critical Exponents, Scaling, and Landau Theory
module: Phase Transitions
moduleNumber: 11
lessonNumber: 4
order: 1104
summary: >
  Near a continuous transition every singular quantity follows a power law in the
  reduced temperature, and the exponents alpha, beta, gamma, delta, nu, and eta
  encode the transition more sharply than T_c itself. Landau theory expands the
  free energy in the order parameter and delivers the mean-field exponents in a
  few lines. They disagree with experiment and with the exact two-dimensional
  Ising values, but the exponents are not independent: the scaling relations of
  Rushbrooke, Widom, Fisher, and Josephson tie them together, and the correlation
  length sets the length scale that organizes universality classes.
topics: [Phase Transitions]
sources:
  - book: Kardar (Statistical Physics of Fields)
    ref: "Ch. 2 — Statistical Fields; §2.6–2.9, and Ch. 3 §3.1–3.3"
  - book: Pathria & Beale
    ref: "Ch. 12 — Phase Transitions; §12.9–12.11"
  - book: Stanley
    ref: "Introduction to Phase Transitions and Critical Phenomena — Ch. 3, Ch. 11–12"
draft: false
---

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**

$$
t = \frac{T - T_c}{T_c},
$$

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:

$$
f(m, T) = f_0(T) + a(T)\, m^2 + b\, m^4 - h m + \cdots ,
$$

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

$$
a(T) = a_0\, t, \qquad a_0 > 0 .
$$

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

$$
2 a\, m + 4 b\, m^3 = h .
$$

$$
% caption: The Landau free energy $f = a(T)m^2 + b\,m^4$; above $T_c$ ($a>0$) a single minimum at $m=0$, at $T_c$ ($a=0$) a flat quartic bottom, and below $T_c$ ($a<0$) two symmetric minima at $\pm m_0$.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (-2.2,0)--(2.2,0) node[right,black]{$m$};
\draw[->,black] (0,-0.4)--(0,3.4) node[above,black]{$f$};
\draw[black!70,very thick,densely dotted] plot[domain=-1.85:1.85,samples=70] (\x,{0.55*\x*\x+0.12*\x*\x*\x*\x});
\draw[black,very thick] plot[domain=-1.85:1.85,samples=70] (\x,{0.12*\x*\x*\x*\x});
\draw[acc,very thick] plot[domain=-1.9:1.9,samples=80] (\x,{0.12*\x*\x*\x*\x-0.55*\x*\x+0.63});
\filldraw[acc] (1.51,{0.12*1.51*1.51*1.51*1.51-0.55*1.51*1.51+0.63}) circle (1.9pt);
\filldraw[acc] (-1.51,{0.12*1.51*1.51*1.51*1.51-0.55*1.51*1.51+0.63}) circle (1.9pt);
\node[black!70,right] at (1.55,2.7) {$T>T_c$};
\node[black,right] at (1.55,1.55) {$T=T_c$};
\node[acc] at (0.72,1.08) {$T<T_c$};
\end{tikzpicture}
$$

## 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 $2am + 4bm^3 = h$ at fixed $T$, $\chi^{-1} =
  2a + 12 b m^2$. 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
  $\alpha = 0$.

$$
% caption: Magnetization versus field for three isotherms; above $T_c$ a straight line of finite slope $\chi$, at $T_c$ the cube-root curve $m\sim h^{1/3}$ with vertical tangent at the origin, and below $T_c$ a discontinuous jump between $\pm m_0$ at zero field.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (-2.4,0)--(2.4,0) node[right,black]{$h$};
\draw[->,black] (0,-1.9)--(0,1.9) node[above,black]{$m$};
% above Tc: linear
\draw[black,very thick,dashed] plot[domain=-1.3:1.3,samples=30] (\x,{0.72*\x});
\node[black,right] at (1.05,0.6) {$T>T_c$};
% at Tc: cube root, parametrized by m
\draw[acc,very thick] plot[domain=-1.55:1.55,samples=60] ({0.45*\x*\x*\x},\x);
\node[acc,above left] at (1.55,1.55) {$T=T_c$};
% below Tc: jump at h=0
\draw[black,very thick] plot[domain=-2.0:-0.05,samples=30] (\x,{-1.0+0.12*\x});
\draw[black,very thick] plot[domain=0.05:2.0,samples=30] (\x,{1.0+0.12*\x});
\draw[black,very thick,dashed] (0,-1.0)--(0,1.0);
\filldraw[black] (0,1.0) circle (1.7pt);
\filldraw[black] (0,-1.0) circle (1.7pt);
\node[black,below right] at (0.05,-1.05) {$T<T_c$};
\end{tikzpicture}
$$

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

$$
\alpha = 0, \quad \beta = \tfrac{1}{2}, \quad \gamma = 1, \quad
\delta = 3, \quad \nu = \tfrac{1}{2}, \quad \eta = 0 .
$$

$$
% caption: Log-log plots near $T_c$ read off the exponents as slopes: $\ln m$ versus $\ln|t|$ has slope $\beta$, and $\ln\chi$ versus $\ln|t|$ has slope $-\gamma$ (the susceptibility diverges as $t\to 0$).
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(6.2,0) node[right,black]{$\ln|t|$};
\draw[->,black] (0,-2.2)--(0,2.2) node[above,black]{};
% ln|t| increases to the right toward 0 at x=5.5 (Tc at far right in -infty... use schematic)
\draw[acc,very thick] (0.6,-1.7) -- (5.4,0.7);
\node[acc,above left] at (5.3,0.7) {magnetization};
\draw[black,very thick,dashed] (0.6,1.8) -- (5.4,-0.9);
\node[black,below left] at (5.3,-0.9) {susceptibility};
\node[black,below] at (5.4,-0.1) {$t\to 0$};
\end{tikzpicture}
$$

## 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.

| Exponent | Mean-field | 2D Ising | 3D Ising | Experiment |
| --- | --- | --- | --- | --- |
| $\alpha$ | $0$ (jump) | $0$ (log) | $0.110$ | $0.11$ |
| $\beta$ | $1/2$ | $1/8$ | $0.326$ | $0.32$–$0.35$ |
| $\gamma$ | $1$ | $7/4$ | $1.237$ | $1.2$–$1.3$ |
| $\delta$ | $3$ | $15$ | $4.79$ | $4.6$–$4.9$ |
| $\nu$ | $1/2$ | $1$ | $0.630$ | $0.62$–$0.65$ |
| $\eta$ | $0$ | $1/4$ | $0.036$ | $0.03$–$0.06$ |

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.[^scaling]

[^scaling]: Kardar (Fields), §4.1, and Stanley, Ch. 11. Widom's homogeneity
ansatz $f_s(t,h) = |t|^{2-\alpha}\,\Phi(h/|t|^{\Delta})$ generates every scaling
relation by matching powers; hyperscaling additionally uses $\xi \sim |t|^{-\nu}$
and that the singular free-energy density scales as $\xi^{-d}$.

- **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 ($0 + 1 + 1 = 2$; $1 = \tfrac{1}{2}\cdot 2$; $1 = \tfrac{1}{2}
\cdot 2$) but satisfies Josephson only at $d = 4$: $\nu d = \tfrac{1}{2}d = 2$
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.

$$
% caption: The correlation length diverges symmetrically as $\xi\sim|t|^{-\nu}$ on both sides of $T_c$; at the critical point fluctuations are correlated on every length scale.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(6.4,0) node[right,black]{$T$};
\draw[->,black] (0,0)--(0,3.9) node[above right,black]{corr. length};
\draw[very thick] (0.4,0.35) .. controls (2.0,0.6) and (2.8,1.3) .. (3.2,2.3);
\draw[very thick] (3.2,2.3) .. controls (3.3,2.9) and (3.36,3.3) .. (3.4,3.6);
\draw[very thick] (3.4,3.6) .. controls (3.44,3.3) and (3.5,2.9) .. (3.6,2.3);
\draw[very thick] (3.6,2.3) .. controls (4.0,1.3) and (4.8,0.6) .. (6.0,0.4);
\draw[acc,dashed] (3.4,0)--(3.4,3.6);
\node[acc,below] at (3.4,0) {$T_c$};
\node[acc,right,align=left] at (4.3,3.0) {power-law\\divergence};
\end{tikzpicture}
$$

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.