6.6 Landau theory

Fundamentals of Landau Theory

Landau theory offers a way to describe phase transitions without needing the microscopic details of a system. Instead of tracking every particle or spin, you write down the most general free energy allowed by the system's symmetries, expressed as a function of a single quantity: the order parameter. Minimizing that free energy then tells you the equilibrium state and the nature of the transition. This approach applies to everything from simple magnets to superconductors, making it one of the most versatile tools in statistical mechanics.

The Order Parameter

The order parameter is the central quantity in Landau theory. It measures how ordered a system is:

• It equals zero in the disordered (high-symmetry) phase.
• It takes a nonzero value in the ordered (broken-symmetry) phase.

What you choose as the order parameter depends on the system. For a ferromagnet, it's the magnetization $M$. For a liquid-gas transition, it's the density difference $\rho_l - \rho_g$. For a superconductor, it's a complex scalar $\psi$ related to the Cooper-pair condensate. The order parameter can be a scalar, a vector, or even a tensor (as in liquid crystals), depending on what kind of order is breaking the symmetry.

Free Energy Expansion

The core idea is to expand the free energy density $f$ as a power series in the order parameter $\phi$:

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

A few key points about this expansion:

• Symmetry dictates which terms appear. If the system has inversion symmetry ($\phi \to -\phi$ leaves the physics unchanged, as in an Ising magnet), all odd powers vanish. If that symmetry is absent, a cubic term $c\,\phi^3$ can appear, which changes the physics significantly.
• The coefficients depend on thermodynamic variables. The crucial assumption is that $a(T)$ changes sign at the transition temperature $T_c$: $a(T) = a_0(T - T_c)$ with $a_0 > 0$. The coefficient $b$ is typically taken as positive and roughly temperature-independent (for a stable theory).
• You truncate the series at the lowest order that captures the essential physics. For a standard second-order transition, keeping terms through $\phi^4$ is sufficient.

You find the equilibrium state by minimizing $f$ with respect to $\phi$: set $\partial f / \partial \phi = 0$ and check that the solution is a minimum.

Symmetry Considerations

Symmetry does a lot of heavy lifting in Landau theory:

• It determines the allowed terms in the expansion (even powers only for $\mathbb{Z}_2$ symmetry, etc.).
• It constrains the form of the order parameter itself.
• It influences whether the transition is continuous or discontinuous. For instance, a cubic invariant generically drives the transition first-order.
• Systems sharing the same order-parameter symmetry and spatial dimensionality fall into the same universality class, meaning they exhibit identical critical behavior.

Phase Transitions in Landau Theory

Second-Order (Continuous) Transitions

For the standard expansion $f = a_0(T - T_c)\phi^2 + b\,\phi^4$ with $b > 0$, minimization gives:

1. Above $T_c$ ($a > 0$): The minimum is at $\phi = 0$. The system is disordered.
2. Below $T_c$ ($a < 0$): The minimum shifts continuously to $\phi = \pm\sqrt{-a/2b} = \pm\sqrt{a_0(T_c - T)/2b}$.

The order parameter grows continuously from zero, there's no latent heat, and the susceptibility $\chi \propto |T - T_c|^{-1}$ diverges at $T_c$. This is the hallmark of a second-order transition.

First-Order (Discontinuous) Transitions

If a $\phi^3$ term is allowed by symmetry, or if $b < 0$ (requiring a stabilizing $\phi^6$ term), the free energy landscape develops competing minima. The order parameter then jumps discontinuously at the transition. First-order transitions feature latent heat, phase coexistence, and hysteresis.

A tricritical point occurs where the coefficient $b$ changes sign, marking the boundary between first-order and second-order behavior in a phase diagram.

Mean-Field Critical Exponents

Landau theory predicts specific power-law exponents near $T_c$. These are called mean-field or classical exponents:

These values are exact for systems above the upper critical dimension ($d = 4$ for scalar order parameters) but differ from experiment in lower dimensions, where fluctuations matter.

Universality Classes

Different physical systems can share the same critical exponents if they have the same:

• Spatial dimensionality $d$
• Symmetry (and dimensionality) of the order parameter

For example, the 3D Ising model (scalar $\phi$, $\mathbb{Z}_2$ symmetry) describes uniaxial magnets and the liquid-gas critical point. The XY model (2-component vector order parameter) describes superfluid helium. This grouping into universality classes means results from a simple model carry over to physically very different systems.

Applications of Landau Theory

Ferromagnetic Systems

The free energy for a ferromagnet with magnetization $M$ in an external field $h$ is:

$f = a_0(T - T_c)M^2 + bM^4 - hM$

• For $h = 0$, this predicts a second-order transition at the Curie temperature $T_c$ with spontaneous magnetization $M \propto (T_c - T)^{1/2}$ below $T_c$.
• The linear $-hM$ term explicitly breaks the $M \to -M$ symmetry, rounding the transition and selecting a preferred direction.

Liquid Crystals

Liquid crystal transitions often involve tensor order parameters that capture orientational order. The isotropic-to-nematic transition, for instance, is typically first-order because the symmetry of the nematic order parameter allows a cubic invariant in the free energy. Landau theory successfully predicts both first-order (isotropic-nematic) and second-order (nematic-smectic A) transitions in these systems.

Superconductivity

Ginzburg and Landau modeled the normal-to-superconducting transition using a complex order parameter $\psi$, where $|\psi|^2$ is proportional to the superfluid density. The free energy expansion, combined with coupling to the electromagnetic vector potential, predicts:

• A second-order transition at the critical temperature $T_c$
• The Meissner effect (expulsion of magnetic flux)
• Flux quantization in units of $\Phi_0 = h/2e$
• The distinction between Type-I and Type-II superconductors

Fluctuations and the Limits of Mean-Field Theory

The Mean-Field Approximation

Landau theory is inherently a mean-field theory: it assumes the order parameter is spatially uniform and ignores fluctuations around the equilibrium value. This works well when:

• Interactions are long-range
• The spatial dimension is high ($d \geq 4$ for scalar order parameters)
• You're not too close to $T_c$

But near the critical point, fluctuations grow and eventually dominate, causing mean-field predictions to fail.

The Ginzburg Criterion

The Ginzburg criterion quantifies how close to $T_c$ you can get before fluctuations invalidate mean-field theory. It compares the magnitude of order-parameter fluctuations to the mean-field value of $\phi$ itself. The result defines a reduced temperature scale, the Ginzburg number $\text{Gi}$, such that mean-field theory breaks down for $|T - T_c|/T_c \lesssim \text{Gi}$.

• For conventional superconductors, $\text{Gi} \sim 10^{-8}$, so mean-field theory works almost all the way to $T_c$.
• For high-$T_c$ superconductors or superfluid helium, $\text{Gi}$ is much larger, and non-mean-field behavior is readily observable.

Correlation Length

The correlation length $\xi$ measures the distance over which fluctuations in $\phi$ are spatially correlated. In Landau theory:

$\xi \propto |T - T_c|^{-\nu}$

with the mean-field exponent $\nu = 1/2$. As $T \to T_c$, $\xi$ diverges, meaning the entire system becomes correlated. This divergence is what makes the critical point so special and is ultimately why mean-field theory fails there: you can't ignore fluctuations when they span the whole system.

Landau-Ginzburg Theory

Spatial Variations of the Order Parameter

To go beyond a uniform order parameter, Landau-Ginzburg theory promotes $\phi$ to a spatially varying field $\phi(\mathbf{r})$ and writes a free energy functional:

$F[\phi] = \int d^d r \left[ a\,\phi^2 + b\,\phi^4 + \kappa\,|\nabla\phi|^2 + \cdots \right]$

The gradient term $\kappa|\nabla\phi|^2$ (with $\kappa > 0$) penalizes spatial variations, encoding the energy cost of inhomogeneity. This extension allows you to describe:

• Domain walls and interfaces between regions of different order
• Topological defects (vortices in superfluids, dislocations in crystals)
• Finite-size effects and the influence of boundary conditions

Coherence Length

The coherence length $\xi_0$ sets the scale over which $\phi(\mathbf{r})$ can vary appreciably. It's determined by the ratio of the gradient stiffness $\kappa$ to the coefficient $a$: roughly $\xi \sim \sqrt{\kappa / |a|}$. Near $T_c$, this diverges with the same exponent as the correlation length. In superconductors, the coherence length governs the size of vortex cores and the spatial profile of the order parameter near surfaces.

Beyond Landau Theory: Scaling and Renormalization

Scaling Hypothesis

Near a continuous transition, thermodynamic quantities obey power laws with universal exponents. The scaling hypothesis goes further: it asserts that the singular part of the free energy is a generalized homogeneous function of its arguments. This leads to:

• Scaling relations connecting different exponents, such as $\alpha + 2\beta + \gamma = 2$ (Rushbrooke) and $\gamma = \beta(\delta - 1)$ (Widom).
• Data collapse: measurements from different systems in the same universality class fall onto a single universal curve when plotted in scaled variables.

These relations are satisfied by experimental data and serve as consistency checks on measured exponents.

Renormalization Group Approach

The renormalization group (RG) provides the theoretical foundation for understanding why universality and scaling work. The basic idea:

1. Coarse-grain the system by integrating out short-wavelength fluctuations.
2. Rescale lengths to restore the original cutoff.
3. Track how the parameters in the free energy (the "couplings") change under this procedure.

The resulting RG flow equations have fixed points that correspond to universality classes. Near a fixed point, the flow determines the critical exponents. The RG explains why mean-field theory works above $d = 4$ (the Gaussian fixed point is stable) and predicts the non-classical exponents observed in $d < 4$.

Experimental Verification

Neutron Scattering

Neutron scattering directly probes the spatial correlations of the order parameter by measuring the structure factor $S(\mathbf{q})$. Near $T_c$, the scattering intensity develops a characteristic Lorentzian peak whose width gives the inverse correlation length $\xi^{-1}$. This technique has been used to measure $\nu$ and $\eta$ (the anomalous dimension exponent) in magnetic systems and to confirm the predictions of RG theory.

Specific Heat Measurements

The specific heat near a phase transition reveals the exponent $\alpha$. Mean-field theory predicts a discontinuous jump ($\alpha = 0$), but experiments on systems like superfluid helium show a logarithmic divergence ($\alpha \approx -0.013$ for the 3D XY universality class), clearly deviating from the Landau prediction. High-resolution calorimetry, including microgravity experiments on the Space Shuttle, has confirmed these non-classical values to remarkable precision.

Extensions and Generalizations

Multicomponent Order Parameters

Many systems have more than one type of order competing or coexisting. Landau theory handles this by introducing multiple coupled order parameters $\phi_1, \phi_2, \ldots$ with cross-coupling terms like $\phi_1^2 \phi_2^2$. This framework predicts:

• Multicritical points (bicritical, tetracritical) where multiple phases meet
• Complex phase diagrams in systems like multiferroics, where magnetic and electric order coexist
• Competing instabilities in unconventional superconductors

Coupling to External Fields

Adding a field $h$ conjugate to the order parameter introduces a term $-h\phi$ in the free energy. This breaks the symmetry explicitly, rounds the transition, and allows you to study crossover behavior. Beyond simple magnetic fields, this framework extends to electric fields, mechanical strain, pressure, and other control parameters, including non-thermal tuning parameters that drive quantum phase transitions at $T = 0$.

Limitations of Landau Theory

Breakdown Near the Critical Point

The central limitation is the neglect of fluctuations. As $T \to T_c$, the correlation length diverges and fluctuations on all length scales become important. The Ginzburg criterion quantifies where this breakdown occurs. Below the upper critical dimension ($d < 4$ for scalar $\phi$), the mean-field exponents are wrong, and you need the renormalization group or numerical simulations to get quantitatively correct results.

Non-Classical Critical Behavior

Real systems in low dimensions show critical exponents that differ significantly from mean-field values. The 2D Ising model, for example, has $\beta = 1/8$ (compared to the mean-field $1/2$) and $\gamma = 7/4$ (compared to $1$). Superfluid helium, liquid-gas critical points, and many magnetic transitions all exhibit non-classical behavior that Landau theory cannot capture quantitatively. Despite this, Landau theory remains the essential starting point: it gives the right qualitative picture, identifies the relevant order parameter and symmetries, and provides the foundation on which more sophisticated methods are built.