= Solution
<Ginzburg-Landau theory of superconductivity> organizes a continuous superconducting transition through a slowly varying complex <order parameter>. In the simplest isotropic single-component SI model,
$$
F-F_n=\int d^3x\left[\alpha|\psi|^2+\frac\beta2|\psi|^4+\frac{|(-i\hbar\nabla-q\mathbf A)\psi|^2}{2m_*}+\frac{|\mathbf B|^2}{2\mu_0}\right],
\qquad \mathbf B=\nabla\times\mathbf A.
$$
Here $\beta>0$, $m_*>0$, $\alpha$ changes sign at $T_c$, and $q=-2e$ fixes a charge/phase convention. The phenomenological normalization of $\psi$ determines the coefficient values; it should not be confused without qualification with a literal density of independently identifiable pairs.
<Microscopic derivation of Ginzburg-Landau theory> expands the BCS response near $T_c$ in small gap amplitude and slow spatial gradients. It identifies the complex order parameter with a scaled pair amplitude, supplies the coefficients from the electronic microscopic parameters, and explains the pair charge in the gauge coupling. The agreement is a controlled near-transition limit, not a statement that the truncated local functional remains exact at all temperatures. <Lev Gor'kov> established this connection explicitly in 1959.<Microscopic Derivation of the Ginzburg-Landau Equations in the Theory of Superconductivity by Lev Gor'kov>[1959 abstract and derivation: near-$T_c$ reduction and twice-electron charge]{ref}
Below the transition, minimization gives $|\psi_0|^2=-\alpha/\beta$ and condensation-energy density $\alpha^2/(2\beta)$. A variation with respect to $\psi^*$ gives
$$
\frac{(-i\hbar\nabla-q\mathbf A)^2}{2m_*}\psi+\alpha\psi+\beta|\psi|^2\psi=0.
$$
Variation with respect to $\mathbf A$ gives the <supercurrent>
$$
\mathbf j=\frac{q\hbar}{m_*}\operatorname{Im}(\psi^*\nabla\psi)-\frac{q^2}{m_*}|\psi|^2\mathbf A,
$$
coupled to the <magnetostatic Ampère law>. For nearly uniform amplitude this reduces to <London screening> and the <Meissner effect>. The theory supplies two different length scales:
$$
\xi^2=\frac{\hbar^2}{2m_*|\alpha|},\qquad
\lambda^2=\frac{m_*}{\mu_0q^2|\psi_0|^2},\qquad \kappa=\frac\lambda\xi.
$$
The amplitude can vary over the <superconducting coherence length>, whereas the <magnetic field> is screened over the <London penetration depth>.
The surface-energy distinction at $\kappa=1/\sqrt2$ explains why type-I and type-II behavior are possible within the same framework. In the type-II regime, quantized winding produces an <Abrikosov vortex> with a finite core and circulating current. Periodic solutions yield an <Abrikosov vortex lattice>. Linearization about the normal state gives $B_{c2}=\Phi_0/(2\pi\xi^2)$, while core and field energetics determine the lower entry field. These predictions connect a spatial field theory to measurable vortex patterns and critical fields, beyond a spatially uniform BCS gap equation. Thin-film critical-field observations helped motivate the large-$\kappa$ regime; magnetic-decoration experiments subsequently made vortex positions visible.<Nobel Lecture: Type-II superconductors and the vortex lattice by Alexei Alexeyevich Abrikosov>[pp.975–977, film motivation, mixed-state solutions and experimental vortex evidence]{ref}
These phenomena also enable applications. Type-II conductors support high-field magnets, but <flux pinning> is needed to limit vortex-motion losses. Phase coherence and fluxoid quantization permit persistent-current loops and interference devices. A weak link supplies <Josephson coupling energy>, allowing SQUID magnetometry and nonlinear quantum circuits; a junction's detailed current–phase law requires a weak-link model, rather than following automatically from the spatially uniform bulk functional. Microstructure can raise a useful critical current without greatly changing the equilibrium upper field, illustrating the difference between thermodynamic stability and transport performance.
The simple functional has identifiable limits. Far below $T_c$, the small-gap expansion is uncontrolled; rapidly varying fields can require nonlocal microscopic electrodynamics. Very near a transition in a fluctuating low-dimensional or strongly anisotropic material, mean-field minimization can miss important thermal or quantum fluctuations. Unconventional or multiband pairing can require several order-parameter components and additional gradient couplings. Strong fields can introduce spin-polarization pair breaking omitted by the orbital model. Finally, the static functional predicts equilibrium extrema, not quasiparticle spectra, dissipation, vortex velocities or a unique amplitude-mode frequency without further dynamical input. These are reasons to choose an appropriate extension or a microscopic calculation, not reasons to discard the successful near-transition predictions.
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