11.3 Landau theory of phase transitions

Landau Theory of Phase Transitions

Fundamentals of Landau theory

Landau theory is a phenomenological approach to phase transitions. Rather than tracking every microscopic interaction in a system, it describes the transition using a single macroscopic quantity called the order parameter. This makes it remarkably general: the same mathematical framework applies to magnets, superconductors, and liquid-gas systems.

The core assumptions are:

• The system is in thermodynamic equilibrium, and the free energy is an analytic function of the order parameter near the critical point.
• The order parameter is small near the transition, so the free energy can be expanded as a Taylor series in powers of the order parameter.
• The equilibrium state corresponds to the minimum of this free energy expansion.

Landau theory works best for continuous (second-order) phase transitions, such as the ferromagnetic transition at the Curie temperature or the onset of superconductivity. It can also be extended to describe first-order transitions with some modifications.

Construction of Landau free energy

The central object is the Landau free energy functional, written as an expansion in the order parameter $\phi$:

$F(\phi) = F_0 + \alpha \phi^2 + \beta \phi^4 + \gamma (\nabla \phi)^2 + \ldots$

• $F_0$ is the free energy of the disordered phase (the baseline).
• $\alpha$, $\beta$, and $\gamma$ are expansion coefficients that depend on temperature and other system parameters.
• The $\gamma (\nabla \phi)^2$ term penalizes spatial variations in the order parameter and becomes important when you study inhomogeneous systems or fluctuations.

Second-order transitions. For systems with a symmetry $\phi \to -\phi$ (like a ferromagnet, where flipping all spins is equivalent), only even powers of $\phi$ appear:

$F(\phi) = F_0 + \alpha(T) \, \phi^2 + \beta \, \phi^4$

The key feature is that $\alpha$ changes sign at the critical temperature $T_c$. A standard choice is $\alpha(T) = a(T - T_c)$ with $a > 0$. Above $T_c$, $\alpha > 0$ and the minimum sits at $\phi = 0$ (disordered). Below $T_c$, $\alpha < 0$ and two new minima appear at nonzero $\phi$. The coefficient $\beta$ must be positive ($\beta > 0$) to keep the free energy bounded from below.

First-order transitions. When the $\phi \to -\phi$ symmetry is absent, a cubic term $\phi^3$ can appear, or equivalently the expansion is carried to sixth order:

$F(\phi) = F_0 + \alpha \, \phi^2 + \beta \, \phi^4 + \delta \, \phi^6$

Here $\beta$ can be negative (unlike the second-order case), and $\delta > 0$ is needed for stability. Because $\alpha$ and $\beta$ change signs at different temperatures, the order parameter jumps discontinuously at the transition rather than growing smoothly from zero.

Order parameters in phase transitions

The order parameter $\phi$ is the physical quantity that distinguishes the ordered phase from the disordered one. It equals zero in the disordered (high-symmetry) phase and takes a nonzero value in the ordered (low-symmetry) phase.

Different systems have different order parameters:

• Ferromagnetic transition: the magnetization $M$. Above the Curie temperature, thermal fluctuations destroy long-range magnetic order and $M = 0$. Below it, spins align and $M \neq 0$.
• Superconducting transition: the superconducting gap (or the complex pair amplitude $\psi$). It's zero in the normal state and nonzero in the superconducting state.
• Liquid-gas transition: the density difference $\rho_l - \rho_g$ between the liquid and gas phases. At the critical point this difference vanishes continuously.

To find the equilibrium value of $\phi$, you minimize the free energy:

$\frac{\partial F}{\partial \phi} = 0$

For the second-order expansion $F = F_0 + \alpha \phi^2 + \beta \phi^4$, this gives:

1. Differentiate: $2\alpha \phi + 4\beta \phi^3 = 0$.
2. Factor: $\phi(2\alpha + 4\beta \phi^2) = 0$.
3. Solutions: $\phi = 0$ (disordered) or $\phi^2 = -\frac{\alpha}{2\beta}$ (ordered, valid only when $\alpha < 0$, i.e., $T < T_c$).

The behavior of $\phi$ near $T_c$ defines the nature of the transition:

• Second-order: $\phi$ grows continuously from zero as $T$ drops below $T_c$, typically as $\phi \propto (T_c - T)^{1/2}$.
• First-order: $\phi$ jumps discontinuously from zero to a finite value at the transition temperature.

Stability analysis near critical points

Knowing which solutions are actual minima (not maxima or saddle points) requires checking the second derivative of the free energy:

• $\frac{\partial^2 F}{\partial \phi^2} > 0$: the phase is stable (a true minimum).
• $\frac{\partial^2 F}{\partial \phi^2} < 0$: the phase is unstable (a maximum or saddle point).

Second-order transitions. Above $T_c$, the only minimum is at $\phi = 0$. Below $T_c$, the curvature at $\phi = 0$ becomes negative, so the disordered phase turns unstable and the system moves to one of the new minima at $\phi \neq 0$.

First-order transitions. The situation is richer because both phases can coexist near the transition temperature:

• The stable phase sits at the global minimum of $F(\phi)$.
• The metastable phase sits at a local minimum. It's not the lowest-energy state, but a finite energy barrier traps the system there temporarily.

The metastable phase can persist until the system reaches a spinodal point:

1. At the spinodal, the local minimum flattens out and the second derivative drops to zero.
2. Beyond the spinodal, the metastable minimum disappears entirely, and the phase becomes unstable.
3. The system then undergoes a rapid, barrier-free transition to the stable phase.

This is why first-order transitions exhibit hysteresis and superheating/supercooling: the metastable phase survives past the equilibrium transition temperature until the spinodal is reached.