Landau Theory and Order Parameters

Landau theory describes continuous phase transitions by expanding the free energy in powers of an order parameter that measures the broken symmetry of the ordered phase.

Definition

Landau theory is a phenomenological description of phase transitions that expands the free energy as a power series in an order parameter respecting the symmetry of the system, with the equilibrium value of the order parameter found by minimizing this free energy.

Scope

This topic covers the concept of an order parameter, the symmetry-constrained Landau expansion of the free energy, the prediction of continuous and first-order transitions according to the sign and structure of the expansion coefficients, the mean-field critical exponents that follow, and the extension to spatially varying order parameters in Ginzburg-Landau theory. The breakdown of mean-field theory below the upper critical dimension is noted.

Core questions

• What is an order parameter and how does it signal spontaneous symmetry breaking?
• How does the symmetry of a system constrain the terms in the Landau free energy?
• How does Landau theory distinguish continuous from first-order transitions?
• Why and where does mean-field Landau theory break down near a critical point?

Key concepts

• Order parameter and broken symmetry
• Landau free-energy expansion
• Spontaneous symmetry breaking
• Mean-field critical exponents
• Ginzburg-Landau theory and the Ginzburg criterion

Key theories

Landau free-energy expansion

Expanding the free energy in even powers of the order parameter, with a coefficient that changes sign at the transition, predicts a continuous onset of order and the classical mean-field critical exponents.

Clinical relevance

Landau theory provides the conceptual language of order parameters and symmetry breaking used across ferromagnetism, ferroelectricity, superconductivity, liquid crystals, and superfluidity, and its field-theoretic extension underlies the Ginzburg-Landau theory of superconductors.

History

Landau introduced the order-parameter expansion in 1937 to give a unified phenomenology of continuous transitions; with Ginzburg he extended it to spatially varying order parameters, yielding the Ginzburg-Landau theory later derived microscopically for superconductors.

Debates

Validity of mean-field exponents

Landau theory predicts universal mean-field critical exponents, but experiments and exact solutions show different values in low dimensions; reconciling this required recognizing the role of fluctuations and the renormalization group beyond mean-field theory.

Key figures

• Lev Landau
• Vitaly Ginzburg

Seminal works

• landaulifshitz1980stat
• goldenfeld1992

Frequently asked questions

What is an order parameter?

It is a quantity that is zero in the disordered, symmetric phase and nonzero in the ordered phase, such as the magnetization of a ferromagnet; its appearance measures and characterizes the symmetry broken at the transition.