= Ginzburg-Landau theory of superconductivity
{c}
<Vitaly Lazarevich Ginzburg> and <Lev Landau> applied an <order parameter> description to <superconductivity> in 1950. A spatially varying complex amplitude extends the fixed-amplitude description of the <London equations>. Their initial charge assumption preceded the later identification of the universal pair charge.<Nobel Lecture: On superconductivity and superfluidity (what I have and have not managed to do) as well as on the physical minimum at the beginning of the XXI century by Vitaly Lazarevich Ginzburg>[pp.983–984, complex order parameter and the charge question]{ref}
Consider the following isotropic, local SI <Helmholtz free energy> model near a continuous <superconducting transition temperature>:
$$
F[\psi,\mathbf A]=\int d^3x\left[
\alpha|\psi|^2+\frac{\beta}{2}|\psi|^4+
\frac{|(-i\hbar\nabla-q\mathbf A)\psi|^2}{2m_*}+
\frac{|\mathbf B|^2}{2\mu_0}\right],\qquad
\mathbf B=\nabla\times\mathbf A.
$$
Here $\beta>0$, $m_*>0$ is a phenomenological mass parameter, and $q=-2e$ fixes the electron-pair convention. The uniform superconducting minimum for $\alpha<0$ has $|\psi_0|^2=-\alpha/\beta$. The gradient term penalizes changes of the amplitude and the gauge-invariant <quantum phase>; the magnetic term penalizes <magnetic field> penetration. The <gauge-coupling sign in Ginzburg-Landau superconductivity> keeps the <electric current> and field equation consistent. This model omits material anisotropy, microscopic nonlocality and critical fluctuations.
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