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 $\phi$.

11.1 Landau Free Energy

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 $f$:

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

11.2 Zero-Field Solutions

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 $\beta = 1/2$.

11.3 Susceptibility

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 $\gamma" = 1$.

11.4 Specific Heat

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).