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In 1986, Johannes Georg Bednorz and Karl Alexander Müller precipitated mixed oxalates from aqueous barium, lanthanum and copper nitrates using oxalic acid. Heating at $900{}^{\circ }\mathrm{C}$ for five hours preceded pressing at $4\mathrm{kbar}$ and reheating for sintering. X-ray diffraction resolved three phases. Rectangular pieces carried gold electrodes and indium wires; four-terminal sensing in a continuous-flow cryostat measured electrical resistance versus temperature. Annealing atmosphere and measuring electric current density were varied; larger electric current densities suppressed the drop. An onset near $35\mathrm{K}$ differed from the much lower temperature where the drop approached three orders of magnitude.^[ref]

This was evidence for possible granular superconductivity, with phase identification unfinished. The paper deferred Meissner effect and heat capacity tests; its proof note corrected bulk lanthanum/barium ratios. A resistive onset did not establish a uniform bulk superconducting transition temperature.^[ref]

Their 1987 authored lecture describes later magnetometer checks on lead, followed by magnetic susceptibility measurements: samples with a resistive drop developed diamagnetism at somewhat lower temperatures. Combining magnetic, transport and X-ray diffraction results identified barium-substituted $\mathrm{L}{\mathrm{a}}_2\mathrm{Cu}{\mathrm{O}}_4$ as the superconducting phase. These later results strengthened the case beyond the initial mixed-phase transport evidence.^[ref]

The Field-cooled Meissner experiment starts above the superconducting transition temperature with a weak, steady applied magnetic field already penetrating a solid sample. Cooling while keeping the applied magnetic field fixed tests whether the interior magnetic flux changes. This is the decisive contrast with cooling in zero magnetic field and switching the magnetic field on afterwards. The diagram shows the two histories for the same final applied magnetic field.

Meissner and Ochsenfeld reported magnetic expulsion in 1933, showing that the transition involves more than vanishing electrical resistance. A modern motivating version appears in MIT's lead-sample experiment: surround the sample with a solenoid supplying a steady magnetic field and a separate pickup coil connected to a recording instrument. Apply the magnetic field while the lead is still normal, then cool through its transition without changing the solenoid electric current. Expulsion produces a transient pickup-coil electric potential difference; warming through the transition produces the opposite sign as magnetic flux returns. MIT also checks the signal direction by switching off the applied magnetic field while the sample is normal. ^{[ref] [ref]}

Faraday's law of induction explains the readout. For a stationary $N$-turn pickup coil, with the same linked magnetic flux $\Phi$ per turn and a fixed orientation,Thus a transient measures a change in linked magnetic flux, not directly the static magnetic field at every point in the sample. For stipulated $N=100$ and $\Delta \Phi =-2\times 10^{-8}\mathrm{Wb}$ per turn, the signed pulse area is $+2\times 10^{-6}\mathrm{V}\mathrm{s}$. The apparatus geometry determines how a change in interior magnetic field contributes to that linked magnetic flux. Keeping the applied magnetic field fixed, repeating on warming, and checking a normal-state control make the expulsion interpretation testable; a stationary pickup coil has no sustained signal after the magnetic flux has stopped changing.

Why does the preparation history matter? In the stationary, ideal perfect conductor model, the interior electric field is zero. The differential form of Faraday's law of induction then givesIf the interior magnetic field was ${\mathbf{B}}_0$ when perfect conduction began, it remains ${\mathbf{B}}_0$; if it was zero, it stays zero when an external magnetic field is subsequently applied. Both a perfect conductor prepared in zero magnetic field and a sample exhibiting the Meissner effect can therefore exclude a newly applied weak magnetic field. Only the field-cooled comparison tests expulsion of an already present magnetic field. This argument concerns a stationary sample and the ideal zero-electric field model, not a moving conductor.

The equilibrium London equations supply the additional constraint. In a homogeneous region without penetrating flux tubes, they imply ${\nabla }^2\mathbf{B}=\mathbf{B}/{\lambda }_L^2$. For a half-space $z\ge 0$ with a magnetic field parallel to its flat boundary, fixed surface value $B_s$, and a bounded solution deep inside, this givesAt $z=5{\lambda }_L$ only $e^{-5}\simeq 0.00674$ of the surface magnetic field remains. The screening electric currents near the surface produce this reduction. Unlike the frozen interior magnetic field of a perfect conductor, this equilibrium profile does not retain an arbitrary initial ${\mathbf{B}}_0$. The extra constitutive relation, and not zero electrical resistance by itself, selects the screened state. Expulsion occurs during the transition, when transient electric fields can be present; the steady-state zero-electric field argument does not describe the process of establishing superconductivity. The diagram gives equilibrium end states, not transition kinetics. ^[ref]

These ideal comparisons concern a solid sample in equilibrium and a sufficiently weak magnetic field. Stronger magnetic fields can destroy superconductivity or permit magnetic flux to penetrate parts of a sample, so complete bulk expulsion is not a claim about every superconducting state. ^[ref] A hole in a superconducting loop can retain magnetic flux even while the material surrounding it screens its own interior; see flux conservation in a superconducting loop. The hole is not superconducting material, so its retained magnetic flux does not contradict the Meissner effect. ^[ref]

Fritz London and Heinz London formulated a phenomenological theory of the magnetic response of superconductors. Their equations make a static magnetic field decay over a finite London penetration depth, explaining field expulsion in the Meissner effect. This adds a physical condition beyond the acceleration of a perfectly conducting electron fluid. The London equations connect a measurable length scale with superconducting currents, providing a useful macroscopic description before a microscopic pairing theory was available.^[ref]

A physicist associated with the London dispersion force and London equations.

Heinz London and Fritz London formulated the London equations in 1935 to describe the magnetic response of a superconductor. Zero electrical resistance alone does not specify expulsion of an existing magnetic field; their additional magnetic relation accounts for the Meissner effect. In a simple planar geometry it gives a field decaying into the material as $B(z)=B(0)e^{-z/{\lambda }_L}$, where ${\lambda }_L$ is the London penetration depth. Their phenomenological theory links a measurable penetration length to superconducting electrodynamics without supplying a microscopic pairing mechanism.^[ref]

The Meissner effect is the equilibrium expulsion of a sufficiently weak, static magnetic field from the bulk of a material exhibiting superconductivity. Expulsion includes a magnetic field that was already present before cooling: zero electrical resistance alone does not predict this. The Field-cooled Meissner experiment tests that distinction. A finite London penetration depth means that the magnetic field and screening electric current occupy a surface layer, rather than disappearing at a mathematical boundary.^[ref]

This makes superconductivity useful for shielding weak magnetic fields, and makes its magnetic response an independent test of the phase. The striking result is that the same screened equilibrium can be reached by two different preparation histories. The assumptions matter: Finite-thickness Meissner screening can leave an appreciable magnetic field throughout a small sample, and the Limits of Meissner screening distinguish quantum vortex penetration and retained magnetic flux from this equilibrium state.

Yoichiro Nambu's superconductivity work led to the spontaneous symmetry breaking ideas recognized by his share of the 2008 Nobel Prize in Physics.^[ref] His 1960 treatment made the Meissner effect response consistent with gauge invariance by including collective pair contributions alongside the quasiparticles. The resulting Ward identity relates the interaction correction to the current and the self-energy; retaining the quasiparticles alone omits part of the response.^[ref]

The individual Nobel lecture explains the bridge to particle: the superconductivity gap suggested a dynamically generated fermion mass, and the accompanying collective excitation suggested a light pion. It also notes that the long-range Coulomb interaction turns the charged phase response into a plasmon. A neutral global phase mode and a charged material therefore need different low-momentum descriptions.^[ref]

In 1961, Yoichiro Nambu and Giovanni Jona-Lasinio supplied a four-fermion model with a self-consistent mass and a massless pseudoscalar bound state: an idealized pion. Their original constituents were nucleon fields. Their introduction explicitly distinguishes the zero-mass neutral collective excitation from its charged plasma counterpart and allows approximate chiral symmetry as the explanation of a nonzero pion mass. This is a model of dynamical symmetry breaking, rather than a microscopic derivation of quantum chromodynamics.^[ref]

In the modern two-light-quark account, chiral symmetry breaking leaves isospin intact and produces three Goldstone bosons in the exact non-anomalous chiral limit. Small quark masses explicitly break that chiral symmetry, so the observed pions are pseudo-Goldstone bosons. The chiral protection of a fermion mass identifies the distinct obstruction to inserting an explicit fermion mass into a chirally symmetric equation. The useful lesson is that massive individual excitations and much lighter collective excitations can coexist because they describe different motions of an interacting state.

A concrete experimental motivation for treating the electromagnetic response carefully is the Field-cooled Meissner experiment. MIT's realization cools lead in a solenoid field using helium gas. A nearby thermometer follows the cooling; a test coil feeds an integrator and oscilloscope. Expulsion gives a pulse, warming reverses it, and switching off the field while the sample is normal checks its sign. A change in alternating-field response alone can reflect normal electrical conductivity. This readout tests magnetic flux expulsion, not the spectrum or existence of a neutral Goldstone boson.^[ref]

For a global symmetry in field theory, different orientations can be physically distinct states. A local gauge symmetry instead includes redundant descriptions of the same physical state. A gauge-dependent field expectation alone is consequently insufficient as a physical criterion for spontaneous symmetry breaking. The Higgs mechanism changes the physical mode content when the gauge field is dynamical; the global Goldstone theorem alone does not prove that mechanism. In particular, the massless direction of a gauge-fixed scalar potential need not appear as a separately observable massless particle.

Consider an original electrical circuit model between two equipotential cross-sections. A normal path has electrical resistance $R_n>0$. Let $b\in \{0,1\}$ indicate whether an additional path connects those same cross-sections; when connected it has electrical resistance $r\ge 0$. Ideal voltage contacts measure those cross-sections. This collapses a spatial network to two parallel branches; it is not a fit to the Bednorz-Müller experiment.

For $r>0$, both branches have the same electric potential difference $V$. Ohm's law and the Kirchhoff node law give $I=V/R_n+bV/r$, henceThe last case follows directly from the ideal short's constraint $V=0$, provided it can carry the imposed electric current below its superconducting critical current. It does not follow by substituting $r=0$ into $V/r$. A normal bridge in series with otherwise superconducting pieces belongs in $r$; superconducting grains alone need not short the measured interval.

Choose $R_n=1\Omega$. A connected bridge with $r=0.01\Omega$ gives $R_{\mathrm{eff}}=1/101\Omega \simeq 9.901\mathrm{m\Omega }$, almost a hundredfold drop despite finite electrical resistance. An ideal connected superconducting filament instead gives zero in this model even when its volume is a small part of the specimen. Isolated superconducting islands can have the same superconducting phase fraction $f$ while leaving $b=0$. For the illustrative disconnected geometry we stipulate that the normal conducting network retains $R_n$.

To plot a chosen connectivity sequence, set $b(f)=0$ for $f<f_c$ and $b(f)=1$ for $f\ge f_c$, with $r=0$ on the connected branch. The two displayed values of $f_c$ describe different chosen architectures. A real percolation calculation would need an explicit network and spatial distribution; the scalar phase fraction $f$ does not determine $b$ by itself.

An instrument with unresolved electric potential difference bounded by $\delta V$ reports only $R_{\mathrm{eff}}\lesssim \delta V/|I|$. For chosen $I=1\mathrm{mA}$ and $\delta V=100\mathrm{nV}$ this floor is $10^{-4}\Omega$, assuming offsets and noise are included in the bound. The figure clips smaller model values to that floor. Increasing $|I|$ improves this formal bound only while Joule heating and the path's superconducting critical current remain controlled. None of these transport statements determines the superconducting phase fraction or establishes complete Meissner effect shielding.

Use SI units and normalize the condensate wavefunction so $|\psi {|}^2=n$. With mechanical momentum operator $-i\hslash \nabla -qA$, the charge current density isFor uniform $n$ and $\psi =\sqrt{n}{e}^{i\vartheta }$, this becomes $j=(nq/m)(\hslash \nabla \vartheta -qA)$.

Under the gauge transformation $A\mapsto A+\nabla \chi$, the wavefunction transforms as $\psi \mapsto e^{iq\chi /\hslash }\psi$, so $\vartheta \mapsto \vartheta +q\chi /\hslash$. The two added gradients cancel in $\hslash \nabla \vartheta -qA$, while $|\psi {|}^2$ stays fixed. Thus the superconducting current is gauge invariant. The same cancellation in the full displayed current proves it without requiring a constant amplitude.

For uniform carrier density in a region with smooth single-valued phase, taking a curl gives the London equation $\nabla \times j=-n{q}^{2}B/m$. In the time-independent setting Ampère's circuital law gives $\nabla \times B={\mu }_{0}j$, and Gauss's law for magnetism gives $\nabla \cdot B=0$. Thereforeso the Helmholtz equation and London penetration depth areThe equation has the modified, exponentially decaying Helmholtz sign. For a planar surface at $x=0$ with material in $x>0$, the bounded solution is $B(x)=B(0)e^{-x/\ell }$; the exponentially growing solution is excluded in the bulk. Hence a macroscopic region far from the surface has negligible field: magnetic flux is expelled from the bulk, the Meissner effect. The literal assertion of zero flux everywhere is the bulk limit: finite penetration remains within distance $\ell$ of a surface. Vortices or flux trapped through a hole require different phase or topology assumptions and are not ruled out by this smooth-phase derivation.

Write the superconducting order parameter as $\psi =\sqrt{n_s}{e}^{i\vartheta }$, with carrier mass $m$. Minimal electromagnetic coupling gives mechanical momentum $\hslash \nabla \vartheta -q\mathbf{A}$, so the current density isIt also follows by substituting $\psi$ into $(q\hslash /m)\mathrm{Im}({\psi }^{\ast }\nabla \psi )-(q^2/m)|\psi {|}^2\mathbf{A}$. In a gauge where the phase is constant, this reduces to the London equation $\mathbf{j}=-n_s{q}^2\mathbf{A}/m$.

Under $\mathbf{A}\mapsto \mathbf{A}+\nabla \chi$, the phase changes by $\vartheta \mapsto \vartheta +q\chi /\hslash$. The two gradient changes cancel, proving gauge invariance of the actual current. A formula omitting the phase term is valid only in its chosen gauge.

For a uniform superconductor without vortices, taking a curl gives $\nabla \times \mathbf{j}=-n_s{q}^2\mathbf{B}/m$. Combine this with the stationary Ampère's law $\nabla \times \mathbf{B}={\mu }_0\mathbf{j}$ and $\nabla \cdot \mathbf{B}=0$ to obtainFor a magnetic field parallel to the planar interface, translation symmetry reduces the equation to ${\mathbf{B}}^{\prime \prime }=\mathbf{B}/{\lambda }_L^2$. Boundedness and decay as $z\to \infty$, together with the tangential boundary value, give the Meissner effectHere ${\lambda }_L$ is the London penetration depth.

The parallel-field assumption is necessary for this one-dimensional Meissner solution. If the printed arbitrary constant ${\mathbf{B}}_0$ has a nonzero normal component, $\nabla \cdot \mathbf{B}=0$ forces that component to be constant across a translation-invariant half-space, whereas the London equation forces it to vanish. Thus no decaying planar Meissner state with that boundary value exists; one must change the geometry or allow a different magnetic state.

In the Landau-Ginzburg theory functional, $B^2/(2{\mu }_0)$ is the magnetic field energy density; the covariant-gradient term is the carrier kinetic energy including phase gradients and electromagnetic coupling. The terms $\alpha |\psi {|}^2+\beta |\psi {|}^4$ give the local condensation energy. With the normalization in the question, $n=|\psi {|}^2$ is the carrier density.

For a uniform field-free equilibrium, minimize $\alpha n+\beta n^2$ over $n\ge 0$. Stability for large $n$ requires $\beta >0$, and a nonzero minimum requires $\alpha <0$. ThereforeThe factor two follows from the question's quartic coefficient $\beta$, rather than a convention using $\beta /2$.

Let $D_A=-i\hslash \nabla -qA$. Under the gauge transformation,Its squared magnitude is unchanged, $|\psi |$ is unchanged, and $B=\nabla \times A$ is unchanged. Thus the full energy has gauge invariance.

For uniform $n$, write $\psi =\sqrt{n}{e}^{i\varphi }$. The kinetic density is $n(\hslash \nabla \varphi -qA{)}^2/(2m)$. Varying $A$ with compactly supported variations and integrating the magnetic term by parts givesAt a minimum its coefficient vanishes, giving the London equationTaking another curl, using $\nabla \cdot B=0$ and a smooth phase so $\nabla \times \nabla \varphi =0$, yieldsFor a planar surface this has the decaying solution $B(x)=B(0)e^{-x/{\lambda }_L}$ inside the material; the growing solution is excluded by boundedness in the bulk. Hence magnetic fields penetrate only a surface layer of thickness ${\lambda }_L$ and vanish deep in the macroscopic bulk, the Meissner effect. This conclusion uses the uniform-density, smooth-phase setting; a vortex has a singular phase and a nonuniform core and is outside those assumptions.

Varying $A_{\nu }$ gives the Maxwell equationsOn the vacuum manifold, choose unitary gauge so that $\varphi =v$. Then $\mathbf{J}=-2e^2{v}^2\mathbf{A}$ and the London equation isFor a static configuration, $\nabla \times \mathbf{B}=\mathbf{J}$ and $\nabla \cdot \mathbf{B}=0$, soThe magnetic field therefore decays exponentially inside the condensate. This is the Meissner effect, with penetration depth equal to the inverse gauge-boson mass.

A closed conducting enclosure redistributes surface charge so that an empty interior is protected from an external static electric field. Grounding fixes a reference and drains deposited charge, but an ideal isolated closed conductor also provides inward electrostatic shielding. Real seams, apertures and cables need careful bonding, shielding and feedthrough filtering. For time-varying fields the relevant wavelength, conductivity, geometry and penetration depth matter; a static argument alone is insufficient. Differential wiring and small pickup-loop areas can complement shielding, while careless multiple ground connections create unwanted current loops.

Ordinary copper or aluminium does not generally exclude a static magnetic field. High-permeability shells instead guide magnetic flux around the protected volume. Their effectiveness depends on shape, continuity, openings and saturation; magnetic history and mechanical stress can worsen performance. Nested shields and degaussing can reduce residual fields. Superconducting shielding can exploit the Meissner effect, but trapped flux, openings and critical-field limits require attention.

At nonzero frequency, eddy currents in conducting shields oppose changing magnetic flux. The skin depth isfor a good conductor in the usual diffusion approximation. Thickness of several skin depths suppresses penetration, but the very-low-frequency limit is weak for a normal metal. High-permeability and conducting layers can therefore complement each other. Active compensation coils are an additional method, not passive shielding. Test performance by applying known fields along relevant axes and measuring the residual field or induced signal over the apparatus bandwidth; room-temperature shielding materials and electronics can themselves add noise.

Use the reduced attractive reduced BCS pairing Hamiltonian with volume V, ${\xi }_k={\epsilon }_k-\mu$ and attraction $\lambda >0$:A specified pairing shell or an ultraviolet renormalization is part of the model. The trace is grand canonical. Insert fermionic coherent states into its time-sliced trace. The resulting Grassmann variables are antiperiodic, and with inverse-energy imaginary time $0\le \tau \le \beta$ the dimensionless action isThis boundary condition is essential for Fermi-Dirac statistics; replacing it by a bosonic periodic trace changes the answer. A complex Hubbard–Stratonovich transformation uses the local identitywith time-slice factors understood. The pairing field renders the fermion action quadratic. For a uniform static field the independent Nambu spinor block $({\psi }_{k\uparrow },{\bar{\psi }}_{-k\downarrow }{)}^T$ hasIts determinant is $-({\omega }_n^2+E_k^2)$. The phase-independent determinant sign cancels under consistent reference normalization. Integrating the fermions gives the BCS pairing-field effective action, ${\mathcal{S}}_{\mathrm{eff}}={\int }_0^{\beta }V|\Delta {|}^2/\lambda d\tau -\mathrm{Tr}\ln {\mathcal{G}}^{-1}$, with the normal-ordering/reference term retained. The fermionic pairing-block determinant normalization uses one independent pairing block for every k; a fully doubled spin-particle-hole trace instead needs one half. The subtracted grand potential of the reduced BCS model is particularly unambiguous:Performing the Matsubara frequency sum yieldsup to a common constant. At zero gap this reduces exactly to the two-spin free-gas result. The Matsubara derivation of the BCS saddle follows by differentiating with respect to ${\Delta }^{\ast }$ and using ${\beta }^{-1}{\sum }_n({\omega }_n^2+E^2{)}^{-1}=\tanh (\beta E/2)/(2E)$ givesThe second equation fixes the chemical potential at specified number. This stationary action is BCS mean-field theory, not an exact integration of pairing-field fluctuations. With constant single-spin density of states ${\nu }_F$ per volume and a symmetric energy cutoff $\Lambda$, weak coupling gives $\Delta (0)\simeq 2\Lambda e^{-1/(\lambda {\nu }_F)}$ and $k_B{T}_c\simeq (2e^{{\gamma }_E}/\pi )\Lambda e^{-1/(\lambda {\nu }_F)}$. The zero-temperature condensation grand-potential density is $-{\nu }_F{\Delta }^2/2$ to leading weak-coupling order; using a spin-summed density would change these displayed factors. The BCS coherence factors are $u_k^2=(1+{\xi }_k/E_k)/2$ and $v_k^2=(1-{\xi }_k/E_k)/2$, and the associated BCS ground state is the normalized product of empty and occupied pair states. ^[ref]

Allowing a slowly varying complex pairing field generates phase stiffness and, near the transition, the Ginzburg-Landau theory of superconductivity. A neutral paired system has a gapless phase sound mode; in a charged bulk superconductor electromagnetic coupling produces the Anderson-Higgs mechanism in a superconductor and the Meissner effect. These collective fluctuations are distinct from the gapped fermionic excitations. A microscopic dynamical calculation is needed to fix collective-mode dispersion and damping; the static saddle alone does not determine them. ^[ref]

For conventional superconductivity, an attractive interaction pairs opposite-spin fermions. A complex pairing Hubbard–Stratonovich transformation introduces a Cooper pair field $\Delta$. The fermionic fields then appear quadratically, giving the pair-field effective action of a superconductorThe trace convention counts each independent spin-pair block once. The uniform saddle is the BCS gap equation; the fermionic excitations have $E_k=\sqrt{{\xi }_k^2+|\Delta {|}^2}$ in the fully gapped case. A slow-field expansion near the transition yields Ginzburg-Landau theory of superconductivity. Normalize its pair field $\Psi$ so $|\Psi {|}^2$ is an effective pair density, with pair mass $m_{\ast }$ and signed charge $q=-2e$:For $a<0$, the uniform minimum has ${\rho }_0=|\Psi {|}^2=-a/b$, with arbitrary constant phase. Writing $\Psi =(\sqrt{{\rho }_0}+h)e^{i\theta }$ separates the amplitude mode $h$ from the phase. The local potential has curvature term $2|a|h^2$ and coherence scale $\xi =\hslash /\sqrt{2m_{\ast }|a|}$. A neutral condensate has a gapless phase Goldstone mode when suitable dynamics is included.

The charged theory is invariant underThe gauge-invariant condensate momentum is $\hslash \nabla \theta -q\mathbf{A}$. Only $\hslash \nabla \theta -q\mathbf{A}$ and $\hslash \dot{\theta }+q\varphi$ enter physical phase dynamics. At nearly constant amplitude, varying $F$ givesFor a smooth phase, taking the curl and using Ampère's law gives ${\nabla }^2\mathbf{B}=\mathbf{B}/{\lambda }^2$: the Meissner effect screens a static bulk magnetic field over the London penetration depth $\lambda$. In a gauge with $\theta =0$, the phase stiffness is a quadratic term in $\mathbf{A}$, the superconducting version of the Higgs mechanism.

To identify the dynamical mode, a simple lossless long-wave superconducting phase-density action isHere ${\chi }_p$ is the pair compressibility coefficient. Eliminating the longitudinal electric potential givesThus the neutral sound mode becomes a gapped longitudinal plasma oscillation; transverse electromagnetic waves acquire ${\omega }^2=c_{\mathrm{vac}}^2{k}^2+c_{\mathrm{vac}}^2/{\lambda }^2$ in this simple bulk model. This redistribution of the phase degree of freedom into electromagnetic collective modes is the Anderson-Higgs mechanism in a superconductor. Local gauge symmetry is a redundancy, not an independently observable symmetry whose breaking creates another massless particle. ^[ref]

The static functional alone does not determine a frequency or damping rate. The displayed mode dispersions require the added dynamical assumptions; real screening, normal carriers, boundaries and dissipation modify them. The amplitude fluctuation is distinct from the phase mode and is not removed by the gauge choice.

Consider a conventional spin-singlet, weak-coupling, three-dimensional superconductor. Set $\hslash =k_B=1$ in the microscopic action and restore electromagnetic units below. An attraction $\lambda >0$ in an energy shell about the Fermi surface gives a Grassmann actionThe attractive sign permits pairing even when the single-particle Fermi gas is otherwise stable. A complex Hubbard–Stratonovich transformation introduces a pair field $\Delta$ proportional at the saddle to $\lambda ⟨{\psi }_{\downarrow }{\psi }_{\uparrow }⟩$. The action is then quadratic in fermions. Integrating them out gives the BCS pairing-field effective actionHere the trace uses one independent Nambu spinor pairing block; a fully doubled particle-hole representation instead carries its corresponding one-half factor. At a uniform saddle the Bogoliubov--de Gennes Hamiltonian can be written asThe negative branch describes the paired vacuum; excitations are Bogoliubov quasiparticles mixing particles and holes. Extremizing the effective action givesThe physical cutoff or a renormalized interaction is part of the microscopic model; an unregulated contact integral must not be used indiscriminately. With constant density of states $N(0)$ per spin and shell half-width ${\Omega }_c$, the zero-temperature weak-coupling solution is ${\Delta }_0\simeq 2{\Omega }_c{e}^{-1/[N(0)\lambda ]}$. The exponentially small scale follows from the logarithmic pairing susceptibility of states near the Fermi surface. For this ideal weak-coupling model, $2{\Delta }_0/(k_B{T}_c)\simeq 3.53$. The microscopic theory relates the gap, pairing and thermodynamic transition.^[ref]

Close to a continuous transition, for fields varying slowly compared with microscopic scales, expand the effective action in powers and gradients of $\Delta$. Its static part is the Ginzburg-Landau theory of superconductivity:The normalization of $\Delta$ fixes the coefficients; here $a(T)$ changes sign at $T_c$. The uniform minimum has $|{\Delta }_0{|}^2=-a/b$ for $a<0$, condensation free-energy density $-a^2/(2b)$, and characteristic coherence-length scale $\xi =\sqrt{K/|a|}$. Above the transition the minimum is at zero. The quartic stabilization and gradient cost are precisely the ingredients of Landau theory for a complex order parameter. This controlled near-transition expansion supplies a bridge between microscopic BCS theory and macroscopic magnetic response; its coefficients and validity depend on the material and fluctuation regime.^[ref]

Write $\Delta =(|{\Delta }_0|+\eta )e^{i\varphi }$. The radial fluctuation has positive local restoring curvature, whereas the phase has no local potential in a neutral theory with global $U(1)$ symmetry. Freezing the amplitude givesThe neutral case supports a gapless collective phase mode. In the charged case the electromagnetic field couples to this phase rigidity. Under $\mathbf{A}\mapsto \mathbf{A}+\nabla \chi$ and $\varphi \mapsto \varphi +q\chi /\hslash$, the displayed combination is invariant. Locally choosing a gauge with constant phase leaves a quadratic term in $\mathbf{A}$. Varying the free energy gives the transverse London current and magnetic screening scale,Combining this relation with static Maxwell equations gives exponential field penetration, the Meissner effect. Single-valuedness of the pair field around a loop with negligible current gives flux in units ${\Phi }_0=2\pi \hslash /|q|=\pi \hslash /e$. These are gauge-invariant physical consequences of the condensate stiffness.

For the dynamics, the neutral phase and density mode is reorganized by the Coulomb interaction and the dynamical electromagnetic field. In a bulk three-dimensional system the long-wave longitudinal mode has a plasma gap, rather than remaining the neutral sound mode. This is the Anderson-Higgs mechanism in a superconductor: a phase coordinate becomes part of the electromagnetic collective response instead of surviving as an independent massless physical particle. Local gauge transformations are redundancies of the description, so a literal breaking of a local redundancy is not an observable. Phase coherence, current response and magnetic screening provide the physical statements.^[ref]

A static Landau functional alone cannot determine that full dynamical spectrum. A two-dimensional film, for example, need not have the same long-wave plasma dispersion as a bulk charged system. Nor does a positive radial curvature automatically guarantee a sharp Higgs amplitude resonance: coupling to quasiparticles and other modes can damp it. The chosen mean-field phase is also a thermodynamic symmetry-breaking description; an isolated finite system can be formulated with definite particle number. The microscopic pairing saddle, the near-transition Landau expansion and the electromagnetic Anderson-Higgs response describe successive scales of the same superconducting state, with distinct assumptions at each step.

A neutral superfluid has a complex order parameter whose phase describes coherent particle transport. For a Bose condensate it is represented by a field amplitude; for a paired Fermi superfluid it is a pair amplitude. In a symmetry-breaking description, selecting its phase breaks the global particle-number $U(1)$ symmetry. A common phase shift leaves the energy unchanged, while a phase gradient costs energy and produces superfluid velocity. Long-wavelength phase oscillations give a gapless Goldstone mode, generally a sound wave when density fluctuations provide the conjugate dynamics. A finite isolated state with exact particle number need not have a nonzero field expectation: the phase-selected language describes the ordered limit and relative-phase coherence.

In a superconductor, the paired order parameter carries charge of magnitude $2e$ and couples to the electromagnetic field. The meaningful spatial phase combination is $\hslash \nabla \theta -q\mathbf{A}$, with signed pair charge $q$. Gauge symmetry is a redundancy, so “breaking local gauge symmetry” is not a literal breaking of a measurable physical symmetry. The gauge-invariant electromagnetic response includes the Meissner effect and a finite London penetration depth. Long-range Coulomb forces move the would-be neutral phase mode to a plasma scale through the Anderson-Higgs mechanism in a superconductor.

Additional symmetries can be broken by particular pairing states. Conventional isotropic spin-singlet pairing need not break spatial rotations or time reversal. A d-wave superconductor has a gap transforming nontrivially under crystal symmetries, and some triplet or multicomponent states can also break spin-rotation or time-reversal symmetries. These additional choices are not required by superconductivity itself. A nonzero pairing amplitude, phase coherence and electromagnetic screening describe related but distinct aspects of the ordered state.

A neutral superfluid can select a phase of its complex order parameter, breaking a global $U(1)$ symmetry. A spatially uniform change of that phase costs no energy. Slowly varying changes therefore produce a gapless Goldstone mode. For a dilute weakly repulsive Bose gas, the Bogoliubov quasiparticle dispersion isso $E_k\simeq \hslash c_{s}k$ at long wavelengths, with $c_s=\sqrt{g_B{n}_0/m_B}$. Density and phase fluctuations are canonically coupled components of this sound branch, rather than two independent positive-frequency modes. Superfluid helium and dilute atomic condensates supply neutral examples, although the dilute-gas formula is not quantitatively valid for strongly interacting liquid helium.

In a superconductor, the pair field is charged and its phase couples to the electromagnetic gauge field. Only combinations such as $\nabla \theta -q\mathbf{A}/\hslash$ are physically meaningful. The Anderson-Higgs mechanism in a superconductor produces electromagnetic screening and removes the neutral sound mode as an independent gapless charged-density oscillation. In a homogeneous three-dimensional continuum, long-range Coulomb forces give a longitudinal plasma gap, approximately ${\omega }^2={\omega }_p^2+c_s^2{k}^2$ at small $k$. The static transverse response gives the Meissner effect andwhen $n_s$ counts superconducting electrons of effective mass $m$. Counting pairs instead uses density $n_s/2$, charge $-2e$ and mass $2m$, giving the same stiffness. The condensate amplitude may also have a gapped Higgs mode, but that amplitude oscillation is distinct from the electromagnetic Higgs mechanism. Local gauge transformations remain a redundancy; this is not spontaneous breaking of an observable local symmetry in the same sense as the neutral global phase symmetry.

A complex order parameter captures both the amplitude and phase rigidity of a superfluid. A local isotropic Ginzburg-Landau theory writes its free-energy functional, to fourth order in amplitude and second order in gradients, asThe uniform minimum has $\Psi =0$ for $\alpha >0$ and $|{\Psi }_0{|}^2=-\alpha /\beta$ for $\alpha <0$. Its phase remains arbitrary under a global $U(1)$ transformation. Minimizing with respect to ${\Psi }^{\ast }$ gives $-K{\nabla }^2\Psi +\alpha \Psi +\beta |\Psi {|}^2\Psi =0$. The phenomenological description specifies the allowed structures and coefficients; microscopic input is needed to identify the normalization of $\Psi$ and calculate those coefficients. A static functional by itself does not fix the dynamical equation or mode frequencies.

For a dilute three-dimensional Bose gas with short-range repulsion and particle mass $m_B$, the microscopic low-energy Hamiltonian isThe interaction parameter is expressed through the low-energy s-wave scattering length $a_s$, rather than an unrenormalized bare delta potential. In a symmetry-selected description, $\Psi =⟨\hat{\psi }⟩$ and $|\Psi {|}^2=n_0$ is condensate density. Equivalently, fixed-number off-diagonal long-range order identifies the macroscopically occupied natural orbital and its amplitude without a nonzero one-point expectation.

Evaluating $\hat{H}-\mu \hat{N}$ in a coherent-field condensate variational state gives the Ginzburg-Landau functional of a dilute Bose gas, with the mean-field microscopic identificationsIts stationary variational equation is the Gross–Pitaevskii equation. The time-dependent variational action additionally contains $i\hslash {\Psi }^{\ast }{\partial }_t\Psi$, givingThis term expresses density-phase conjugacy. It is the missing microscopic dynamical input; a purely dissipative relaxation law would describe a different approximation.

Bogoliubov theory writes $\hat{\psi }=\Psi +\delta \hat{\psi }$ and retains quadratic fluctuations. In a uniform condensate, $\mu =g_B{n}_0$ and the fluctuation matrix isIts eigenvalues are $\pm \sqrt{{ϵ}_k({ϵ}_k+2g_B{n}_0)}$. The positive branch is the gapless sound mode, continuously crossing over to particle-like excitations at larger $k$. The same result follows by linearizing the time-dependent Gross–Pitaevskii equation about its uniform solution. Thus the microscopic condensate amplitude, scattering interaction and mass determine the mean-field static coefficients and the collective mode. Quadratic quantum fluctuations also produce quantum depletion of a Bose condensate, which a pure coherent-state product omits.

These identifications are controlled for a dilute weakly interacting gas away from a fluctuation-dominated critical region. They are not a universal statement that the exact finite-temperature GL coefficient is always $-\mu$ or that its quartic coefficient never receives corrections. Integrating out noncondensed and short-wavelength modes produces effective temperature-dependent coefficients and can change their near-critical behavior. The phase rigidity described by $K$ and the thermodynamic order-parameter expansion remain the shared organizing structure.

For BCS theory, the microscopic order parameter is instead an anomalous pair amplitude, commonly normalized as an energy gap $\Delta \propto ⟨{\hat{\psi }}_{\downarrow }{\hat{\psi }}_{\uparrow }⟩$. It is not an ordinary single-electron field expectation or automatically the square root of a pair number density. Integrating out fermions and expanding in small $\Delta$ and slow gradients near $T_c$ gives the microscopic derivation of Ginzburg-Landau theory. The BCS pairing-field effective action supplies its coefficients through normal-state pair-response functions and thermal fermion sums.

For example, a clean isotropic three-dimensional weak-coupling s-wave model, with $N(0)$ the density of states per spin at the Fermi energy and $\Delta$ measured in energy units, giveswith microscopic Ginzburg-Landau coefficients from BCS theoryThe numerical factors arise from fermionic Matsubara sums and the isotropic velocity average; they depend on the stated gap and density-of-states conventions. Rescaling the order parameter rescales all three coefficients. The expansion applies near the transition with sufficiently slow spatial variation; disorder, anisotropic pairing and strong coupling alter the coefficients or require additional components. Microscopic derivation also fixes the pair charge in the covariant gradient, producing the superconducting screening current and the Meissner effect.^[ref]

The neutral Bose and charged Fermi theories therefore share a complex order parameter and a phase-stiffness functional, while deriving that field from different microscopic objects. The Bogoliubov transformation mixes particle and hole fluctuations of the bosonic condensate in one case and paired fermionic excitations in the other. The static GL limit organizes long-distance behavior; the full microscopic theories retain excitation spectra and temperature dependence beyond that local near-transition expansion.

The superconducting order is a coherent pair amplitude, schematically $\Delta \propto ⟨c_{\downarrow }{c}_{\uparrow }⟩=|\Delta |e^{i\varphi }$. Under a global fermion phase change $c\mapsto e^{i\alpha }c$, it acquires phase $2\alpha$. A symmetry-broken thermodynamic description selects a pair phase. This must be distinguished from electromagnetic gauge freedom: a local gauge transformation changes the phase and potentials together and is a redundancy of the description, not an observable symmetry that the material literally violates.

A conventional centrosymmetric s-wave superconductor usually has an even-parity orbital pair state and a spin singlet state, consistent with the antisymmetry of two fermions. Its ordinary zero-field equilibrium state preserves time reversal. The electromagnetic coupling makes phase dynamics and screening part of the Anderson-Higgs mechanism in a superconductor; the Meissner effect is a measurable consequence. The condensate phase description does not imply that electric charge is no longer conserved.

In an unconventional superconductor, the gap can transform nontrivially under the crystal point group or have other nonstandard pairing structure. A common example is d-wave superconductivity, with sign changes and possible nodes in the gap. This does not automatically imply an additional observable fourfold-rotation breaking: a rotation that reverses the gap's sign can be accompanied by a gauge phase change. A nematic superconducting state instead breaks an actual crystal rotation symmetry in its gauge-invariant responses.

A chiral combination such as $p_x+i{p}_y$ can break time reversal, while a spin-triplet order parameter can also distinguish spin directions when the applicable spin-rotation and spin-orbit constraints permit it. Helium-3 superfluidity illustrates paired-fermion order with richer spin and orbital structure. These possibilities must be tested rather than equated: unconventional pairing does not by itself mean spin-triplet pairing, time-reversal breaking, or nematicity. Gap angular dependence, phase-sensitive interference, spin response and spontaneous magnetic signatures probe different symmetry questions. The phase-sensitive test of d-wave pairing is more specific than inferring a gap sign merely from its magnitude.

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,Here $\beta >0$, $m_{\ast }>0$, $\alpha$ 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.^[ref]

Below the transition, minimization gives $|{\psi }_0{|}^2=-\alpha /\beta$ and condensation-energy density ${\alpha }^2/(2\beta )$. A variation with respect to ${\psi }^{\ast }$ givesVariation with respect to $\mathbf{A}$ gives the supercurrentcoupled 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: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 $\kappa =1/\sqrt{2}$ 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.^[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.