4.1 Landau theory

Fundamentals of Landau theory

Landau theory describes phase transitions by expanding the free energy as a function of an order parameter near the transition point. Rather than tracking every microscopic degree of freedom, it relies on symmetry to determine which terms can appear in this expansion. This makes it remarkably general: the same framework applies to magnets, superconductors, liquid crystals, and many other systems.

Free energy expansion

The central idea is to write the free energy $F$ as a power series in the order parameter $\phi$:

$F(\phi) = F_0 + a\,\phi^2 + b\,\phi^4 + \cdots$

• You truncate the expansion, keeping only the lowest-order terms allowed by symmetry.
• The coefficients ($a$, $b$, etc.) depend on temperature and other control parameters. Typically $a$ changes sign at the transition: $a = a_0(T - T_c)$.
• The equilibrium state is found by minimizing $F$ with respect to $\phi$. Setting $\partial F / \partial \phi = 0$ gives you the equilibrium order parameter.
• Higher-order terms ($\phi^6$, etc.) are usually negligible far from special points but become important near tricritical points or when lower-order coefficients vanish.

Order parameter concept

The order parameter $\phi$ quantifies how much symmetry has been broken. It equals zero in the disordered (high-temperature) phase and takes a nonzero value in the ordered (low-temperature) phase.

• Magnetization $\vec{M}$ in a ferromagnet: zero above the Curie temperature, nonzero below.
• Density difference $\rho_l - \rho_g$ in a liquid-gas transition.
• Complex gap function $\Delta$ in a superconductor.

Depending on the system, the order parameter can be a scalar ($\phi$), a vector ($\vec{M}$), or a tensor ($Q_{ij}$, as in liquid crystals). Its symmetry properties dictate the entire structure of the Landau expansion.

Symmetry considerations

Symmetry is what makes Landau theory predictive without microscopic details.

• The free energy must be invariant under all symmetry operations of the high-temperature (disordered) phase. For a system with $\phi \to -\phi$ symmetry (like an Ising magnet), only even powers of $\phi$ are allowed.
• If the symmetry permits odd powers (like a cubic term $\phi^3$), the transition is generically first-order.
• Systems that share the same order-parameter symmetry and spatial dimensionality fall into the same universality class, which is why Landau theory explains universal critical behavior across very different physical systems.

Phase transitions in Landau theory

Landau theory classifies transitions by how the order parameter behaves at the transition temperature. The key distinction is whether $\phi$ changes continuously or jumps.

Second-order (continuous) transitions

In a second-order transition, the order parameter grows smoothly from zero as you cool below $T_c$. The free energy contains only even powers of $\phi$ (enforced by symmetry):

$F = F_0 + a_0(T - T_c)\,\phi^2 + b\,\phi^4$

with $b > 0$ for stability. Minimizing gives:

• For $T > T_c$: the minimum is at $\phi = 0$ (disordered phase).
• For $T < T_c$: the minimum shifts to $\phi = \pm\sqrt{\frac{a_0(T_c - T)}{2b}}$.

This yields the power-law behavior $\phi \propto (T_c - T)^{1/2}$, so the mean-field exponent is $\beta = 1/2$. The susceptibility diverges as $\chi \propto |T - T_c|^{-1}$, giving $\gamma = 1$. Examples include ferromagnetic transitions and the superconducting transition in zero field.

First-order transitions

When symmetry allows a cubic term ($c\,\phi^3$) in the expansion, or when the quartic coefficient $b$ is negative (requiring a positive $\phi^6$ term for stability), the transition becomes first-order.

• The order parameter jumps discontinuously at $T_c$.
• There is latent heat and phase coexistence at the transition.
• Metastable states can persist beyond the equilibrium transition, producing hysteresis.

Examples include many liquid-gas transitions and certain structural transitions in crystals (like some martensitic transformations).

Tricritical points

A tricritical point occurs where a line of second-order transitions meets a line of first-order transitions in a phase diagram. At this point, the quartic coefficient $b$ passes through zero, so the $\phi^6$ term controls stability:

$F = F_0 + a\,\phi^2 + b\,\phi^4 + c\,\phi^6$

with $b = 0$ at the tricritical point and $c > 0$. The mean-field tricritical exponents differ from ordinary critical exponents: $\beta = 1/4$ and $\gamma = 1$. Physical examples include the tricritical point in $^3\text{He}$-$^4\text{He}$ mixtures and certain metamagnetic systems.

Critical exponents

Near a continuous phase transition, physical quantities follow power laws in $|T - T_c|$. The exponents characterizing these power laws are called critical exponents, and they provide the most direct way to compare theory with experiment.

Mean-field approximation

Landau theory is a mean-field theory: it assumes the order parameter is spatially uniform and ignores fluctuations. This gives a set of universal mean-field critical exponents:

Universality classes

Systems with the same spatial dimensionality $d$ and order-parameter symmetry (characterized by the number of components $n$) share identical critical exponents. Some important classes:

• Ising ($n = 1$): uniaxial magnets, liquid-gas transitions. In $d = 3$: $\beta \approx 0.326$, $\gamma \approx 1.237$.
• XY ($n = 2$): planar magnets, superfluid helium. In $d = 3$: $\beta \approx 0.348$.
• Heisenberg ($n = 3$): isotropic magnets. In $d = 3$: $\beta \approx 0.366$.

This explains why a ferromagnet and a binary fluid can have the same critical exponents despite being completely different physical systems.

Scaling relations

The critical exponents are not all independent. They satisfy exact identities derived from the homogeneity (scaling) of the free energy near $T_c$:

• Rushbrooke's identity: $\alpha + 2\beta + \gamma = 2$
• Widom's identity: $\gamma = \beta(\delta - 1)$
• Josephson's (hyperscaling) relation: $2 - \alpha = d\,\nu$

These relations reduce the number of independent exponents and provide consistency checks. If your measured exponents don't satisfy these relations, something is off with the measurements or the assumed universality class.

Applications of Landau theory

Ferromagnetic transitions

For a ferromagnet with magnetization $\vec{M}$ in an external field $\vec{H}$, the Landau free energy is:

$F = F_0 + a(T - T_c)M^2 + bM^4 - \vec{H} \cdot \vec{M}$

Minimizing with respect to $M$ gives the equation of state. Above $T_c$, the susceptibility follows $\chi = 1/[2a(T - T_c)]$ (Curie-Weiss law). Below $T_c$, spontaneous magnetization appears. The theory also provides a starting point for understanding magnetic domains, though domain structure requires the spatially extended Ginzburg-Landau version.

Superconducting transitions

The order parameter is the complex gap function $\Delta = |\Delta|\,e^{i\varphi}$, and the free energy includes gradient and magnetic field coupling terms:

$F = \int d^3r \left[ a|\Delta|^2 + b|\Delta|^4 + K|(\nabla - i\frac{2e}{\hbar c}\vec{A})\Delta|^2 + \frac{B^2}{8\pi} \right]$

This predicts two characteristic length scales: the coherence length $\xi$ (over which $\Delta$ varies) and the penetration depth $\lambda$ (over which magnetic fields decay). Their ratio $\kappa = \lambda/\xi$ determines whether the superconductor is type-I ($\kappa < 1/\sqrt{2}$) or type-II ($\kappa > 1/\sqrt{2}$), which controls whether magnetic flux is fully expelled or can penetrate in quantized vortices.

Structural phase transitions

In structural transitions, the order parameter is often an atomic displacement or a strain component. Examples:

• Ferroelectric transitions (e.g., $\text{BaTiO}_3$): the order parameter is the electric polarization $\vec{P}$.
• Martensitic transformations: the order parameter is a strain tensor component.

Landau theory predicts the appearance of soft modes (phonon frequencies that drop toward zero as $T \to T_c$) and can describe coupling between multiple order parameters in multiferroic materials, where magnetic and electric orders coexist and interact.

Limitations of Landau theory

Fluctuations near the critical point

Landau theory treats the order parameter as spatially uniform, but near $T_c$, fluctuations in $\phi$ grow in both amplitude and spatial extent. The Ginzburg criterion quantifies when fluctuations become too large to ignore. It compares the magnitude of fluctuations within a correlation volume to the mean-field order parameter:

$\text{Gi} = \frac{k_B T_c}{b\,\xi_0^d} \ll 1 \quad \text{(for mean-field to hold)}$

where $\xi_0$ is the bare correlation length. When this ratio is not small, mean-field predictions break down.

Breakdown of the mean-field approach

Mean-field theory becomes exact only above the upper critical dimension $d_c = 4$ (for standard short-range interactions). Below $d_c$, fluctuations alter the critical exponents:

• In $d = 3$, deviations from mean-field values are moderate but measurable.
• In $d = 2$, deviations are large. The exact solution of the 2D Ising model gives $\beta = 1/8$, far from the mean-field value of $1/2$.
• In $d = 1$, many systems have no finite-temperature phase transition at all.

Beyond Landau theory

The renormalization group (RG) provides a systematic way to incorporate fluctuations. RG methods explain why universality classes exist, calculate non-classical critical exponents via the $\epsilon$-expansion (where $\epsilon = 4 - d$), and reveal the structure of fixed points governing critical behavior. Conformal field theory gives exact results in $d = 2$. These advanced methods build directly on the Landau framework but go beyond it by treating fluctuations on all length scales.

Ginzburg-Landau theory

Ginzburg-Landau (GL) theory generalizes Landau theory to spatially varying order parameters. This is essential whenever interfaces, defects, or external fields create inhomogeneity.

Extension to spatially varying systems

The free energy becomes a functional that integrates over space:

$F[\phi] = \int d^3r \left[ f(\phi) + K(\nabla\phi)^2 \right]$

The gradient term $K(\nabla\phi)^2$ penalizes rapid spatial variations, so the order parameter prefers to vary smoothly. This functional allows you to describe:

• Domain walls and interfaces between coexisting phases
• Topological defects (vortices, dislocations)
• Finite-size effects and the influence of boundary conditions

Minimizing the functional with respect to $\phi(\vec{r})$ gives the Euler-Lagrange equation, which determines the spatial profile of the order parameter.

Coherence length

The coherence length $\xi$ sets the scale over which the order parameter can vary spatially:

$\xi = \sqrt{\frac{K}{|a|}} \propto |T - T_c|^{-\nu}$

with the mean-field exponent $\nu = 1/2$. As $T \to T_c$, $\xi$ diverges, meaning fluctuations become correlated over arbitrarily large distances. The coherence length also determines the thickness of domain walls and enters the Ginzburg criterion for the validity of mean-field theory.

Critical fields in superconductors

GL theory is especially powerful for superconductors in magnetic fields. It predicts:

• Thermodynamic critical field $H_c$: the field at which the condensation energy equals the magnetic energy.
• Lower critical field $H_{c1}$: the field at which it becomes energetically favorable for magnetic flux to enter a type-II superconductor as quantized vortices.
• Upper critical field $H_{c2} = \Phi_0 / (2\pi\xi^2)$: the field at which superconductivity is completely suppressed.

Between $H_{c1}$ and $H_{c2}$, type-II superconductors enter the mixed state, where magnetic flux penetrates as an Abrikosov vortex lattice. GL theory provides the framework for calculating vortex structure, interactions, and the phase diagram of the mixed state.

Experimental validation

Neutron scattering techniques

Neutron scattering directly probes spatial correlations of the order parameter by measuring the structure factor $S(\vec{q})$. Near a phase transition:

• $S(\vec{q})$ develops a peak that sharpens and grows as $T \to T_c$, reflecting the diverging correlation length.
• The peak width gives $\xi^{-1}$, allowing direct measurement of the correlation length exponent $\nu$.
• In structural transitions, neutron scattering reveals soft phonon modes whose frequencies approach zero at the transition.
• Inelastic neutron scattering measures the dynamic susceptibility $\chi(\vec{q}, \omega)$, giving access to critical dynamics.

Specific heat measurements

The specific heat $C$ near a continuous transition exhibits a singularity characterized by the exponent $\alpha$:

$C \propto |T - T_c|^{-\alpha}$

• Mean-field theory predicts a discontinuous jump ($\alpha = 0$), while real systems often show a divergence or a cusp.
• The famous lambda transition in liquid $^4\text{He}$ at 2.17 K shows a logarithmic singularity ($\alpha \approx -0.013$), clearly deviating from mean-field predictions.
• For first-order transitions, calorimetry detects the latent heat directly.

Susceptibility studies

Measuring the response to a conjugate field (magnetic susceptibility, dielectric susceptibility, compressibility) gives the exponent $\gamma$. Susceptibility measurements also reveal:

• Crossover phenomena between different universality classes as temperature or other parameters change.
• Critical slowing down in dynamic susceptibility, where the relaxation time diverges as $\tau \propto \xi^z$ near $T_c$.

Computational methods

Monte Carlo simulations

Monte Carlo methods simulate statistical mechanical systems by stochastically sampling configurations weighted by their Boltzmann factors. Near phase transitions:

• Algorithms like the Wolff or Swendsen-Wang cluster methods reduce critical slowing down, which plagues simple single-spin-flip (Metropolis) algorithms near $T_c$.
• Simulations on finite lattices of various sizes $L$ are combined with finite-size scaling to extract critical exponents and $T_c$.
• Monte Carlo provides some of the most precise numerical estimates of critical exponents (e.g., 3D Ising: $\nu = 0.6300(4)$).

Renormalization group approach

The RG provides the theoretical foundation for universality and scaling. The key ideas:

1. Coarse-grain the system by integrating out short-wavelength fluctuations.
2. Rescale lengths to restore the original lattice spacing.
3. Identify fixed points of this transformation. Systems flowing to the same fixed point share the same critical exponents.

The $\epsilon$-expansion ($\epsilon = 4 - d$) allows perturbative calculation of critical exponents. For the 3D Ising model ($\epsilon = 1$), RG gives exponents in good agreement with Monte Carlo and experiment.

Finite-size scaling

Real simulations and experiments deal with finite systems. Finite-size scaling theory relates the behavior of a system of size $L$ to the thermodynamic limit through scaling functions:

$\phi(T, L) = L^{-\beta/\nu}\,\tilde{\phi}\left( (T - T_c)\,L^{1/\nu} \right)$

By simulating multiple system sizes and collapsing the data onto a single scaling function, you can extract $T_c$, $\beta/\nu$, and $1/\nu$ simultaneously. This technique is one of the most reliable ways to determine critical parameters from numerical data.