Ginzburg–Landau Theory

Ginzburg–Landau theory is a phenomenological theory of superconductivity formulated by Vitaly Ginzburg and Lev Landau in 1950. It describes the superconducting state through a complex order parameter and a free-energy functional that includes its spatial variation and interaction with a magnetic field. The theory explains magnetic screening, superconducting interfaces, and the distinction between type-I and type-II superconductors without initially specifying a microscopic pairing mechanism. More broadly, the name denotes an order-parameter framework for studying phase transitions. (nobelprize.org)

Order parameter and microscopic interpretation

The superconducting order parameter is a complex field,

\psi(\mathbf r)=|\psi(\mathbf r)|e^{i\theta(\mathbf r)}.

Its magnitude characterizes local superconducting order, while its phase enters the description of supercurrents. With an appropriate normalization, $|\psi|^2$ represents a superconducting condensate density. It is a collective field rather than an ordinary single-particle wave function. (nobelprize.org)

The theory preceded BCS theory, introduced in 1957. In 1959, Lev Gor’kov derived the Ginzburg–Landau equations from BCS theory near the transition temperature, establishing their microscopic basis for conventional superconductors. This derivation identified the charge entering the electromagnetic coupling with the charge of a Cooper pair, of magnitude $2e$ , where $e$ is the elementary charge. (nobelprize.org)

Free-energy functional

For an isotropic, single-component superconductor, a standard expression in SI units is

F[\psi,\mathbf A]=F_n+\int d^3r\, \left[ \alpha|\psi|^2+\frac{\beta}{2}|\psi|^4 +\frac{1}{2m^*} \left|(-i\hbar\nabla-q\mathbf A)\psi\right|^2 +\frac{|\mathbf B|^2}{2\mu_0} \right].

Here $F_n$ is the normal-state reference free energy, $\mathbf A$ is the magnetic vector potential, $\mathbf B=\nabla\times\mathbf A$ , $q=-2e$ , and $m^*$ is a phenomenological mass parameter. The reduced Planck constant is $\hbar$ , and $\mu_0$ is the vacuum permeability. For a prescribed applied magnetic field, the corresponding Gibbs free energy is used. (pcteserver.mi.infn.it)

Near a continuous transition, one usually takes

\alpha(T)\simeq a(T-T_c),\qquad a>0,\quad\beta>0.

In a uniform, zero-field state, minimization gives

|\psi_0|^2= \begin{cases} 0,&T>T_c,\\[2mm] -\alpha/\beta,&T<T_c. \end{cases}

The quartic term stabilizes the free energy, while the gradient term penalizes spatial variation. Its electromagnetic form preserves local gauge invariance, connecting the theory with gauge theory. (pcteserver.mi.infn.it)

Ginzburg–Landau equations

Variation with respect to $\psi^*$ gives

\frac{1}{2m^*}(-i\hbar\nabla-q\mathbf A)^2\psi +\alpha\psi+\beta|\psi|^2\psi=0.

Variation with respect to $\mathbf A$ supplies the superconducting current and its coupling to Maxwell’s equations:

\mathbf j_s= \frac{q}{m^*} \operatorname{Re} \left[\psi^*(-i\hbar\nabla-q\mathbf A)\psi\right].

These coupled nonlinear equations, supplemented by boundary conditions, determine equilibrium order-parameter and magnetic-field configurations. They describe effects absent from a model that assumes a spatially fixed condensate density, including suppressed superconductivity near surfaces and vortex cores. (pcteserver.mi.infn.it)

Characteristic lengths and magnetic classification

Two characteristic lengths govern the simplest theory:

The coherence length $\xi$ sets the scale over which the order parameter varies.
The magnetic penetration depth $\lambda$ sets the screening scale for a weak magnetic field in the bulk superconducting state.

Their ratio,

\kappa=\frac{\lambda}{\xi},

is the Ginzburg–Landau parameter. In the isotropic, single-component theory, $\kappa<1/\sqrt2$ identifies type-I behavior, while $\kappa>1/\sqrt2$ identifies type-II behavior. The boundary corresponds to a change in the sign of the normal–superconducting interfacial energy. (nobelprize.org)

Type-I superconductors exhibit bulk magnetic exclusion through the Meissner effect below their thermodynamic critical field. Type-II superconductors admit magnetic flux as vortices between lower and upper critical fields, $H_{c1}$ and $H_{c2}$ . The simple $\kappa$ classification applies to the conventional model; more elaborate order parameters can produce additional magnetic behavior. (nobelprize.org)

Vortices and flux quantization

An Abrikosov vortex has a core in which superconducting order is suppressed. Around the core, the phase winds by an integer multiple of $2\pi$ , and circulating currents support magnetic flux. An isolated, singly quantized vortex carries one magnetic flux quantum,

\Phi_0=\frac{h}{2e}.

This relates flux quantization to the charge of the paired condensate. (arxiv.org)

In 1957, Alexei Abrikosov used the theory to describe the vortex state of type-II superconductors. In the ideal isotropic bulk model, vortices form a triangular lattice. Its existence allows superconducting order and magnetic flux to coexist over a finite field range. (nobelprize.org)

Broader framework and limitations

Beyond superconductivity, Ginzburg–Landau models expand a coarse-grained free energy in an order parameter and its gradients. Depending on the system’s symmetry, the field may be real, complex, or multicomponent. Applications include ferromagnetism, superfluidity, and nonequilibrium pattern formation. Time-dependent extensions describe relaxation or evolving amplitudes; they require dynamical assumptions beyond the equilibrium functional. (arxiv.org)

The conventional superconducting expansion is controlled near $T_c$ , where the order parameter is small and varies slowly. Minimizing the functional alone is a mean-field approximation: it does not fully account for thermal fluctuations near a critical point. The Ginzburg criterion estimates when those fluctuations invalidate mean-field predictions. Treating the field statistically and using the renormalization group extends the framework to critical scaling and universality classes. (pcteserver.mi.infn.it)

References

Advanced information on the Nobel Prize in Physics, 7 October 2003nobelprize.org
Vitaly L. Ginzburg — Nobel Lecturenobelprize.org
The Ginzburg-Landau theory of superconductivity (1950)pcteserver.mi.infn.it
An introduction to the Ginzburg-Landau theory of phase transitions and nonequilibrium patternsarxiv.org
The Ginzburg-Landau Theory of Type II superconductors in magnetic fieldarxiv.org