Ginzburg-Landau theory

A long-wavelength order-parameter description extending a local Landau free energy by gradient terms. For a one-component complex field, a common isotropic form is $\mathcal{F}=\int [\alpha |\Psi {|}^2+\beta |\Psi {|}^4/2+K|\nabla \Psi {|}^2]d^{3}r$. A charged field instead uses a covariant gradient. Static minimization determines equilibrium profiles; a dynamical law requires additional input. Ginzburg-Landau theory of superconductivity and the Ginzburg-Landau functional of a dilute Bose gas realize this structure with different microscopic normalizations.

Take $\Psi =⟨\hat{\psi }⟩$ normalized so that $|\Psi {|}^2$ is condensate density. The coherent-field expectation of a dilute contact-interacting Bose Hamiltonian gives the grand-potential functional $\int [{\hslash }^2|\nabla \Psi {|}^2/(2m_B)+(V_{\mathrm{ext}}-\mu )|\Psi {|}^2+g_B|\Psi {|}^4/2]d^{3}r$. Its leading microscopic coefficients are therefore those displayed, with positive s-wave scattering length $a_s$ for repulsion. Varying with respect to ${\Psi }^{\ast }$ gives the stationary Gross–Pitaevskii equation. Thermal and quantum fluctuations can renormalize the coefficients; these mean-field identities are not exact universal critical coefficients.