Landau Theory of Phase Transitions

Landau theory provides a phenomenological framework for continuous (second-order) phase transitions by expanding the free energy in powers of an order parameter $\phi$.

The Landau free energy density (in the absence of external fields) is:

$f(\phi, T) = f_0(T) + \frac{1}{2}a(T)\phi^2 + \frac{1}{4}b\phi^4 + \frac{1}{6}c\phi^6 + \cdots$

Assumptions:

• $f$ is analytic in $\phi$ near the transition
• Symmetry $\phi \to -\phi$ (e.g., Ising systems) eliminates odd powers
• $b > 0$ for stability
• $a(T) = a_0(T - T_c)$ changes sign at $T_c$

With an external field $h$ conjugate to $\phi$Add $-h\phi$:

$f(\phi, T) = f_0 + \frac{1}{2}a(T)\phi^2 + \frac{1}{4}b\phi^4 - h\phi$

The equilibrium order parameter minimizes $f$:

$\frac{\partial f}{\partial \phi} = a\phi + b\phi^3 - h = 0$

For $h = 0$:

• $T > T_c$ ($a > 0$): minimum at $\phi = 0$ (disordered phase)
• $T < T_c$ ($a < 0$): minima at $\phi = \pm\sqrt{-a/b} = \pm\sqrt{a_0(T_c - T)/b}$

The order parameter grows as:

$\phi = \begin{cases} 0 & T > T_c \\ \pm\sqrt{a_0(T_c - T)/b} & T < T_c \end{cases}$

This yields the mean-field critical exponent $\beta = 1/2$.

The susceptibility $\chi = \partial\phi/\partial h|_{h=0}$ is obtained by expanding $\phi(h) = \phi_0 + \chi h + \cdots$:

$a\phi + b\phi^3 - h = 0 \implies (a + 3b\phi_0^2)\chi = 1$

• $T > T_c$: $\phi_0 = 0$So $\chi = 1/a = 1/[a_0(T - T_c)]$Giving $\gamma = 1$.
• $T < T_c$: $\phi_0^2 = -a/b$So $\chi = 1/(-2a) = 1/[2a_0(T_c - T)]$Giving $\gamma" = 1$.

The free energy at equilibrium is:

$f_{\text{eq} = \begin{cases} f_0 & T > T_c \\ f_0 - a^2/(4b) & T < T_c \end{cases}}$

The specific heat discontinuity is:

$C_{T_c^-} - C_{T_c^+} = -T_c \frac{\partial^2}{\partial T^2}\left(\frac{-a^2}{4b}\right)\bigg|_{T_c} = \frac{T_c a_0^2}{2b}$

This is a finite jump ($\alpha = 0$ in mean-field theory).

• Critical exponents (mean-field): $\beta = 1/2$ (order parameter), $\gamma = 1$ (susceptibility), $\alpha = 0$ (specific heat jump), $\delta = 3$ (critical isotherm: $h \propto \phi^3$ at $T = T_c$).
• Universality: Systems with the same symmetry and dimensionality share the same critical exponents, regardless of microscopic details. Landau theory gives mean-field exponents, which are exact only above the upper critical dimension ($d > 4$ for short-range interactions).
• Clausius-Clapeyron analogue: At a first-order transition (when $b < 0$), the discontinuity in the order parameter gives a latent heat $L = T_c \Delta s$ where $\Delta s = -\partial f/\partial T$ is the entropy jump.
• Ginzburg criterion: Mean-field theory is valid when fluctuations are small, i.e., when $|T - T_c| > T_c(a_0^2 k_B^2)/(32\pi^2 b^2 \xi_0^d)$. For $d < 4$, this fails very close to $T_c$.

• Assuming Landau theory is always valid: It is a mean-field theory. Near $T_c$ in low dimensions, critical fluctuations dominate and renormalisation group methods are required.
• Forgetting that $b$ can be negative: If $b < 0$, the $\phi^6$ term must be included to ensure stability. The transition becomes first-order, and the simple $\phi = \pm\sqrt{-a/b}$ solution does not apply.
• Neglecting the role of symmetry: The form of the Landau expansion depends on the symmetry of the order parameter. A vector order parameter (e.g., in the XY model) requires a different expansion than a scalar.
• Confusing the order parameter with a physical observable: The order parameter $\phi$ is an abstract quantity. For a ferromagnet it is the magnetisation; for a superfluid it is the condensate wavefunction; for a liquid-gas transition it is the density difference.

• Ferromagnetic transitions: The Landau theory with $\phi = M$ (magnetisation) predicts the Curie temperature and the Curie-Weiss law $\chi \propto 1/(T - T_c)$ for the susceptibility above $T_c$.
• Superfluid helium: The order parameter is the complex condensate wavefunction $\psi$. The Landau-Ginzburg expansion includes $|\psi|^2$ terms and gradient terms, leading to the Ginzburg-Landau theory of superconductivity.
• Binary alloys: The order parameter describes the degree of chemical ordering (e.g., Cu-Zn ordering in brass). The Landau theory predicts the order-disorder transition temperature.
• Liquid crystals: Nematic-isotropic transitions can be described by a tensor order parameter $Q_{ij}$. The Landau expansion includes both scalar and tensor invariants.

A magnetic system has Landau coefficients $a(T) = 0.5(T - 400)$ K and $b = 2.0$ (arbitrary units). Find $T_c$ and the magnetisation at $T = 300$ K.

At $T_c$, the coefficient $a(T_c) = 0$, so $0.5(T_c - 400) = 0$ giving $T_c = 400$ K.

At $T = 300$ K: $a(300) = 0.5(300 - 400) = -50$. The equilibrium magnetisation is:

$M = \sqrt{-a/b} = \sqrt{50/2.0} = \sqrt{25} = 5.0$

The free energy at equilibrium: $f_{\text{eq}} = -a^2/(4b) = -2500/(8) = -312.5$.

The susceptibility above $T_c$: $\chi = 1/a = 1/[0.5(T - 400)]$. At $T = 500$ K, $\chi = 1/50 = 0.02$.

The specific heat jump at $T_c$: $\Delta C = T_c a_0^2/(2b) = 400 \times 0.25/(4) = 25$.