Landau theory
/ LAN-dow /
Faced with a phase transition, Lev Landau made a brilliantly economical guess: near the transition the order parameter is small, so simply expand the free energy as a power series in it, keeping only the terms that the system's symmetry allows. Without solving any microscopic model, this humble Taylor expansion reproduces the qualitative shape of the transition — the appearance of order, its temperature dependence, the response to an applied field. It is the universal grammar of phase transitions.
For a scalar order parameter M with the Ising symmetry M to -M (so only even powers appear), the free energy near the transition is F(M) = F_0 + a(T) M^2 + b M^4 + ..., with b > 0 for stability and a(T) = a_0 (T - T_c) changing sign at T_c. Minimizing F over M tells the whole story: above T_c, a > 0 and the only minimum is M = 0 (disordered); below T_c, a < 0 and new minima appear at M = plus or minus sqrt(-a / 2b), which is proportional to (T_c - T)^(1/2). Landau theory thus predicts the mean-field critical exponents beta = 1/2, gamma = 1, alpha = 0 (a jump in specific heat), and delta = 3. Adding a gradient term (K/2)(grad M)^2 to allow the order parameter to vary in space gives the Landau-Ginzburg functional, the starting point for domain walls, interfaces, and superconductivity.
Landau theory is the shared language of order parameters, symmetry breaking, and phase transitions, and it works quantitatively in the right circumstances. But it is essential to be honest: it is a MEAN-FIELD theory that ignores fluctuations of the order parameter, so its critical exponents are simply wrong below the upper critical dimension (four for the Ising class), where fluctuations dominate. The Ginzburg criterion quantifies the temperature window near T_c in which Landau theory fails. It works well precisely where fluctuations are weak — conventional superconductors, with their very long coherence length, and any system above four dimensions.
Minimizing F = a(T) M^2 + b M^4 gives M = 0 above T_c and M is proportional to (T_c - T)^(1/2) below it, so Landau theory predicts beta = 1/2. Experiment on a 3D magnet gives beta about 0.33 instead: the honest gap between mean-field theory and reality that the renormalization group later closed.
Landau theory gives the qualitative transition but the mean-field exponent beta = 1/2, not the true value.
Landau theory is mean-field: it neglects order-parameter fluctuations, so its exponents (beta = 1/2, etc.) are wrong below four dimensions. It is not merely an approximation to be refined but qualitatively incomplete near a critical point in three dimensions, which is exactly what the renormalization group repairs.