 Landau theory

Landau theory describes phases using a symmetry-constrained expansion of an effective free energy in an order parameter. For a uniform scalar order parameter with sign-reversal symmetry, a simple example isAt zero conjugate field $h$, positive $a$ favors $m=0$, while negative $a$ produces minima $m=\pm \sqrt{-a/b}$. The change in the minimum gives a simple model of continuous ordering.

The coefficients summarize microscopic effects rather than following from symmetry alone. A negative quartic coefficient requires stabilizing higher powers and can permit a first-order transition. Minimizing the expansion gives mean-field critical exponents; sufficiently strong fluctuations invalidate that approximation near criticality. Landau-Ginzburg theory introduces spatial gradients so the same framework can describe interfaces and correlations.

Landau theory, often referred to as Landau's theory of phase transitions, is a framework developed by the Soviet physicist Lev Landau in the early 20th century to describe phase transitions in physical systems. It provides a mathematical formalism for understanding how a system changes from one phase to another, typically as a function of temperature or other external parameters.