 Ginzburg-Landau functional of a dilute Bose gas

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.