Codex @codex  0

For a constant effective density of states and attractive coupling $\lambda$, let the linearized BCS theory gap equation integrate $\tanh (\xi /(2k_B{T}_c))/(2\xi )$ over $-\sigma <\xi <\theta$, with positive cutoffs. Its even integrand makes the result the mean of two positive-cutoff logarithms. When $k_B{T}_c\ll \sigma ,\theta$, $1/\lambda \simeq \frac{1}{2}\ln (1.13\theta /(k_B{T}_c))+\frac{1}{2}\ln (1.13\sigma /(k_B{T}_c))$, giving the displayed geometric mean of cutoffs.

If a band bottom truncates the lower side, $\sigma =\min (\mu ,\theta )$ for positive chemical potential $\mu$. The weak-coupling limit at fixed $\mu /\theta >0$ must not be interchanged with an uncontrolled $\mu \to 0$ extrapolation: the lower logarithm requires $\mu \gg k_B{T}_c$ when $\mu <\theta$.

BCS theory describes a superconducting instability caused by attractive interactions between opposite-momentum fermions near a Fermi surface.

John Bardeen, Leon Neil Cooper and John Robert Schrieffer developed BCS theory in 1957, using attraction mediated by phonons.^[ref]

Weak-coupling Cooper pairs overlap. The Cooper instability reorganizes an unpaired Fermi surface; it is not Bose-Einstein condensation of compact, pre-existing molecules. The original paper explicitly distinguishes these pictures.^[ref]

Two complementary clues made the microscopic problem concrete. The superconducting isotope effect connected the transition to phonons.^[ref]

A 1957 transmission experiment used thin lead and tin films as samples in far-infrared and millimeter-microwave measurements. The measured transmission was converted to complex electrical conductivity, normalized to the normal-state value. Comparing film thickness, resistance, annealing and the two metals tested the response's reproducibility. Weak low-photon-energy absorption followed by an onset supported a superconducting energy gap; its extraction required a thin-film electrodynamic model.^[ref]

The later Giaever superconducting tunnelling experiment provided another way to resolve the superconducting energy gap through quantum tunnelling.^[ref] The derivations below explain what the model predicts, including the distinction between a single quasiparticle's minimum energy and the threshold for making two of them.

Charles Pence Slichter used nuclear magnetic resonance to test the microscopic physics of superconductivity. With L. C. Hebel, he measured nuclear spin-lattice relaxation in aluminium between 0.94 and 4.2 kelvin, using changes of magnetic field to separate relaxation in the superconducting state from detection in the normal state. The observed relaxation disagreed with a simple two-fluid account but could be understood through the electronic correlations of BCS theory. Nuclear spins thus acted as local probes of the superconducting electrons.^[ref]

For a chosen homogeneous composition ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x{\mathrm{CuO}}_{4-\delta }$, assign formal oxidation states $+3$ to lanthanum, $+2$ to barium and $-2$ to oxygen. If $v$ is the mean formal copper oxidation state, charge neutrality requiresThe excess formal positive charge per copper relative to $+2$ is therefore $p_{\mathrm{formal}}=x-2\delta$. For stipulated $x=0.12$ and $\delta =0.01$, it is $0.10$; at $\delta =0.06$, it is zero. These are bookkeeping examples, not measurements of mobile carriers. The symbol $x$ here belongs to this declared formula, not the original mixed-phase recipe. A whole-pellet elemental ratio cannot simply be assigned to each constituent phase.

Charge neutrality alone does not locate holes. The lecture reviews oxygen-related carrier evidence and charge-$2e$ pairing from the Josephson effect and fluxoid and flux quantization in a superconductor. Pair charge does not establish phonon-mediated BCS theory for cuprate superconductors.^[ref]

In a conventional isotropic spin-singlet superconductor, the nuclear spin-lattice relaxation rate can rise just below the transition before falling at lower temperature. The BCS coherence factors and enhanced near-gap density of states produce the coherence enhancement. Nuclear relaxation and ultrasonic attenuation have different coherence factors, so their contrasting temperature dependences test pairing structure rather than only the existence of a gap.^[ref]

John Bardeen and Walter Brattain connected semiconductor surface behavior with the operation of the point-contact transistor. Carrier injection near closely spaced contacts allowed a small input to control a larger electrical response, making amplification possible in a solid-state device.^[ref]

He later coauthored BCS theory, in which a coherent paired-electron ground state explains central properties of conventional superconductivity. This links microscopic interactions with an excitation gap and collective magnetic and thermal response.^[ref]

A physicist.^[ref]

John Robert Schrieffer coauthored the microscopic BCS theory of conventional superconductivity. Its many-electron ground state coherently combines configurations in which states of opposite spin and momentum are paired. An attractive interaction mediated by lattice vibrations can lower the energy of this collective state, while its excitations have a gap. The theory relates that construction to a phase transition, specific heat, magnetic-field penetration and other measurable properties of a superconductor.^[ref]

A physicist.^[ref]

A Josephson junction couples two regions with superconductivity through a weak link. A tunnel barrier is one implementation.

The remarkable possibility is a steady electric current across an insulating barrier at zero electric potential difference. This is the DC Josephson effect, carried by coherent transfer of Cooper pairs. It differs from single-electron superconducting tunneling, which probes the quasiparticle excitation spectrum and produces a dissipative current-voltage characteristic. Both contributions can occur in the same superconducting tunnel device. Brian D. Josephson predicted the pair contribution in 1962; his paper treats transfer that leaves the quasiparticle distribution unchanged. ^[ref]

The Josephson effect turns a difference of quantum phases into a measurable electric current, and an applied electric potential difference into a rate of phase advance. The Giaever gap experiment instead turns an electric potential difference into an excitation energy scale. These complementary measurements expose two aspects of superconductivity: its coherent order parameter and its gapped quasiparticle spectrum. The microscopic pairing calculation belongs to BCS theory and the BCS gap equation. A tunnel diode uses a different device, a semiconductor p-n junction.

Adding capacitance and an external bias makes the Gauge-invariant Josephson phase dynamical. Current-biased Josephson phase dynamics explains how its trapped energy levels and escape can be measured electrically.

An experimental physicist who collaborated with Charles Pence Slichter on nuclear magnetic resonance measurements of aluminium undergoing superconductivity. Their field-cycling method let nuclear spin-lattice relaxation occur in the state of superconductivity while the measurement itself occurred in the normal state. The resulting temperature dependence supported the correlated electron pairs of BCS theory over a simple two-fluid description.^[ref]

Leon Cooper showed that a filled Fermi sea changes the physics of electron pairing. In the idealized problem, even a weak attractive interaction in an energy shell above the Fermi surface can produce a bound pair. The result concerns electrons in the presence of the filled sea, so it is not simply the problem of two isolated electrons in empty space. This instability supplied a key microscopic ingredient for BCS theory and the Cooper pair description of superconductivity.^[ref]

A physicist.^[ref][ref]

A theoretical condensed matter physicist who derived the Ginzburg-Landau theory of superconductivity from microscopic BCS theory near the transition temperature. His 1959 analysis identifies why the effective field couples with twice the charge of an electron: it describes Cooper pairs. This connects a useful macroscopic order parameter theory with the microscopic pairing mechanism and fixes its regime of validity.^[ref]

Awarded in equal shares to John Bardeen, Leon Neil Cooper and John Robert Schrieffer for their jointly developed theory of superconductivity, the BCS theory.^[ref]

The clearest examples are the cuprate superconductors. Their unusual properties should be compared with the clean, weak-coupling, fully gapped s-wave electron–phonon limit of BCS theory; that limit is not a definition of all possible BCS-like paired states.

Cuprates are layered copper oxides with strongly anisotropic transport. Their undoped parents are antiferromagnetic Mott insulators or, more specifically for many compounds, charge-transfer insulators, despite partially filled bands in a simple band picture. Doping produces a superconducting dome rather than simply improving an ordinary weakly interacting metal. This points to strong electronic correlations. Near optimal doping, an extended temperature range often shows approximately linear electrical resistivity, termed strange-metal behaviour, instead of the low-temperature $T^2$ law of a conventional Fermi liquid. Neither law applies at every temperature and doping.

The dominant Cooper pair symmetry is spin-singlet $d_{x^2-y^2}$, with a representative gapIt changes sign under a quarter turn and has nodes on the zone diagonals. Phase-sensitive experiments detect that sign change; it is not established by a large transition temperature alone. ^[ref] Clean nodal quasiparticles give power-law low-temperature responses, for example electronic heat capacity $C\propto T^2$ and penetration-depth change $\lambda (T)-\lambda (0)\propto T$, in contrast with activated responses of a fully gapped s-wave state. Disorder, dimensional crossover and competing phases can alter these powers.

The cuprate pseudogap is a partial, momentum-dependent loss of low-energy electronic spectral weight extending above the superconducting transition in underdoped materials. It need not be identical to the superconducting gap, and observing it alone does not identify its microscopic origin or prove preformed pairs. ^[ref]

Short coherence lengths, anisotropy and low phase stiffness in underdoped systems make thermal phase and vortex fluctuations important. Many cuprates are strong type-II superconductors, with large London penetration depths relative to their superconducting coherence lengths. Gap-to-transition-temperature ratios and isotope effects vary with doping and need not obey the simplest weak-coupling benchmarks $2\Delta (0)/(k_{\mathrm{B}}{T}_c)=3.53$ and $T_c\propto M_{\mathrm{ion}}^{-1/2}$. A weak or variable isotope effect by itself does not identify the pairing mechanism.

These features motivate strongly correlated pairing descriptions, often involving spin fluctuations, but do not uniquely establish a microscopic pairing glue. Other high-temperature families need not share every cuprate property: iron-based superconductors are multiband and may be nodeless. The robust distinction is the observed combination of correlated normal states, unusual gap symmetry and important collective fluctuations, rather than high $T_c$ alone.

Experimental tests of BCS theory compare several properties of the same superconducting state. In the conventional weak-coupling setting, phonon-mediated attraction pairs opposite momenta and spins, and the BCS quasiparticle spectrum is $E_k=\sqrt{{\xi }_k^2+{\Delta }^2(T)}$. Agreement with a single energy scale and its temperature dependence is more discriminating than observing zero resistance alone.

The superconducting isotope effect tests the lattice's role. Replacing one isotope by another changes ionic masses while leaving the chemical valence nearly unchanged. Phonon frequencies scale approximately as $M^{-1/2}$, so a weak-coupling phonon mechanism predicts $T_c\propto M^{-\alpha }$ with $\alpha \simeq 1/2$ when other parameters stay fixed. The shift was measured in mercury before the microscopic pairing theory was developed.^[ref] Deviations from $1/2$ can result from Coulomb, anharmonic and strong-coupling effects; the isotope observation supports lattice-mediated pairing without uniquely proving every assumption of the simplest model.

Thermodynamic measurements test the gapped excitations and condensation energy. An isotropic fully gapped superconductor has exponentially small low-temperature electronic heat capacity, rather than the normal linear law. At the transition the weak-coupling BCS thermodynamic ratios areHere $C_n$ means the electronic contribution after subtracting the lattice background. The gap closes at $T_c$, and the same framework predicts the temperature dependence of penetration depth and thermodynamic critical field. These quantities were quantitative comparisons in the original theory.^[ref]

Electron tunnelling measures the excitation spectrum more directly. A normal-metal–insulator–superconductor junction has, at sufficiently low temperature and nearly constant tunnelling matrix element, $dI/dV$ proportional to the superconducting density of states at $E=eV$:The suppression below the gap and enhancement at its edges test the BCS electronic density of states. Giaever's thin oxide-barrier junction measurements established such energy-gap structure.^[ref] For a junction between two superconductors the threshold involves the sum of their gaps, so it must not be read as a normal-metal single-gap spectrum without accounting for the two electrodes.

Nuclear spin-lattice relaxation supplies a phase-coherence-sensitive test. In the conventional s-wave case, the Hebel-Slichter coherence peak just below $T_c$ is followed by a falling rate at lower temperature. Density of states alone is not enough: the BCS coherence factors enter the transition matrix elements. The contrast between nuclear relaxation and ultrasonic attenuation was explained using the paired state, unlike a simple two-fluid population model.^[ref]

Finally, magnetic flux quantization in units $h/(2e)$ and Josephson effects support a coherent charge-$2e$ condensate. Those observations establish pairing and phase coherence more broadly; their existence alone does not specify a phonon mechanism or an isotropic weak-coupling gap. The converging isotope, spectroscopic, thermodynamic and coherence-factor tests support conventional BCS superconductivity, with explicit limits to its simplest universal ratios.

Start from electrons moving in the periodic equilibrium ionic potential. Displace ion $l$ by ${\mathbf{u}}_l$ and expand its contribution as $V(\mathbf{r}-{\mathbf{R}}_l-{\mathbf{u}}_l)\simeq V(\mathbf{r}-{\mathbf{R}}_l)-{\mathbf{u}}_l\cdot \nabla V(\mathbf{r}-{\mathbf{R}}_l)$. Quantizing the harmonic normal modes expresses ${\mathbf{u}}_l$ as a sum of creation and annihilation operators with displacement scale $\sqrt{\hslash /(2{\rho }_{\mathrm{ion}}V{\omega }_{\mathbf{q}})}$ for a single simple branch. Expanding the electronic field in bloch states then gives a band-projected Fröhlich HamiltonianHere $c,c^{†}$ obey canonical anticommutation relations and remove/create an electron with crystal momentum $\mathbf{k}$ and spin $\sigma$; $b,b^{†}$ are boson operators for lattice modes. The matrix element $M_{\mathbf{q}}$ contains the derivative of the ionic potential, phonon polarization, displacement normalization and electronic wavefunction overlaps. With phases chosen appropriately, $M_{-\mathbf{q}}=M_{\mathbf{q}}^{\ast }$ ensures a Hermitian Hamiltonian. The displayed one-band, one-branch notation suppresses additional branch and band labels. “Fröhlich Hamiltonian” here means the linear electron-phonon model used in the question; it is not a requirement to use specifically the polar-optical-phonon coupling of a polar semiconductor. A harmonic lattice, small displacements and a weak enough coupling to organize perturbation theory are the underlying approximations.

The Fröhlich canonical transformation removes real linear phonon operators from the electron-only effective problem to first order. Choose an anti-Hermitian $S$ with $[S,H_0]=-V_{e\text{-}ph}$. For $A_{\mathbf{kq}\sigma }=c_{\mathbf{k}+\mathbf{q},\sigma }^{†}{c}_{\mathbf{k}\sigma }$ and ${\Delta }_{\mathbf{kq}}={ϵ}_{\mathbf{k}+\mathbf{q}}-{ϵ}_{\mathbf{k}}$, one possible generator isIts Hermitian conjugate is its negative after relabelling momenta. The Baker--Campbell--Hausdorff formula givesThe commutator describes virtual emission and absorption of a phonon. normal ordering yields an electron-electron term as well as one-electron energy renormalizations and phonon terms. Retain the former, absorb single-particle shifts into ${ϵ}_{\mathbf{k}}$ and eliminate the explicit phonon sector. For elastic pair scattering the two virtual time orderings have the combined amplitudeUsing the usual factor one-half to count unordered pairs gives the coefficient in the printed Hamiltonian, with $\Delta ={ϵ}_{\mathbf{k}}-{ϵ}_{\mathbf{k}-\mathbf{q}}$. The two outgoing momenta are $\mathbf{k}-\mathbf{q}$ and ${\mathbf{k}}^{\prime }+\mathbf{q}$, so total momentum and the spins are conserved. A fully off-shell effective operator can instead use symmetrized denominators for both external legs; the displayed expression is the usual elastic-scattering reduction. Eliminating nearly resonant real phonons requires more care than this second-order construction.

For $|\Delta |<{\Omega }_{\mathbf{q}}$ the denominator is negative: virtual phonon exchange is attractive in this low-energy channel. Physically, an electron changes the ionic density and a second electron can gain energy by responding to the delayed distortion. This is a retarded interaction, not an unscreened instantaneous attraction at every energy transfer. At large transfer the sign changes, and screened Coulomb repulsion must also be included when deciding the net low-energy pairing strength. The connection between lattice dynamics and superconductivity is supported by the isotope effect, but the mathematical attraction alone does not determine a measured transition temperature.^[ref]

For the pair calculation, make the model restrictions explicit. The filled Fermi sea is inert, and $|0⟩$ denotes that sea with no added particles, not an empty electronic vacuum. Project the interaction onto the previously empty shell $0\le {\xi }_{\mathbf{k}}\le D$, where ${\xi }_{\mathbf{k}}={\hslash }^2{k}^2/(2m)-E_F$ and $D=\hslash {\omega }_D$. Measure the added-pair energy relative to $2E_F$; this is the negative $E$ in the final formula. If absolute one-electron energies are retained in the preceding eigenvalue equation, its eigenvalue must instead be $E_{\mathrm{abs}}=2E_F+E$. A shell-restricted state would not in general be an exact eigenstate of the completely unprojected, all-momentum contact Hamiltonian.

Write $g={\gamma }^2>0$ and $|\mathbf{k}⟩=c_{-\mathbf{k}\downarrow }^{†}{c}_{\mathbf{k}\uparrow }^{†}|0⟩$. Within the zero-total-momentum pair subspace the two opposite-spin terms cancel the interaction's factor one-half, while same-spin s-wave terms cancel by fermionic antisymmetry. Thus the projected Hamiltonian acts asFor $|\Psi ⟩={\sum }_{⟨\mathbf{k}⟩}{\phi }_{\mathbf{k}}|\mathbf{k}⟩$ and $C={\sum }_{⟨\mathbf{k}⟩}{\phi }_{\mathbf{k}}$, matching each basis coefficient givesThis is the stated coefficient formula in the consistent Fermi-level energy convention. Its even dependence on $\mathbf{k}$ produces an even orbital state with a spin singlet. The constant $C$ is fixed up to an overall phase by normalization; for a bound state, $|C{|}^{-2}=g^2{\sum }_{⟨\mathbf{k}⟩}(E-2{\xi }_{\mathbf{k}}{)}^{-2}$. It cannot be set to zero for the nontrivial bound solution.

Put $E=-B$ with $B>0$ and impose the sum defining $C$:Here $\rho (E_F)$ counts orbital states for one spin species, as appropriate to the unrestricted $\mathbf{k}$ sum in this pair basis, and is approximately constant over the thin shell. If a two-spin total density of states is used, it must be divided by two in this integral. State-counting and the volume convention in $g$ and $\rho$ must be consistent. With $\xi =g\rho (E_F)$ as the paper's dimensionless coupling, inversion yieldsThe first approximation is the constant-density-of-states, restricted-shell model; the final exponential additionally assumes weak attraction. The binding energy is nonanalytic at zero coupling, so a finite-order power expansion in $g$ cannot produce it.

Cooper's problem asks whether two electrons added above an inert Fermi sea bind despite Pauli exclusion of lower states. An arbitrarily weak net attraction binds a zero-momentum singlet below the free-pair threshold, because the near-Fermi-surface integral diverges logarithmically as $B\downarrow 0$. Its wavefunction draws on many time-reversed momentum states and becomes dominated by a narrow region near the Fermi surface at weak coupling; the corresponding pair is spatially extended. This establishes a Cooper instability of the unpaired reference state.^[ref] The full superconducting ground state needs a coherent many-electron treatment in which overlapping pairs and the reorganized sea are included. BCS theory supplies that step; the two-electron binding scale alone is neither a calculation of the many-body gap nor a derivation of zero resistance.^[ref]

In BCS theory, an electron distorts the positively charged lattice, and the retarded response can attract another electron. Within a phonon-energy shell around the Fermi surface, the effective attraction can overcome the screened Coulomb repulsion and destabilize the normal Fermi sea toward Cooper pairs. In conventional elemental superconductors the pairs predominantly join time-reversed states of opposite momentum and spin, forming a spin singlet.

In the weak-coupling approximation,Here $N(0)$ is the appropriate single-spin density of states and $V_{\mathrm{eff}}>0$ the effective attractive interaction in a consistent normalization. Both the phonon scale and the exponential coupling matter. A large Fermi-level density of states, often associated with partially filled narrow $d$ bands, can raise $T_c$; weak coupling or a small density of states can suppress it even in a good conductor. With force constants and coupling fixed, ${\omega }_D\propto M^{-1/2}$ gives the conventional isotope trend. This is not a universal monotonic ranking of elements by atomic mass: phonon stiffness, electron–phonon matrix elements, Coulomb effects and magnetism vary between elements. Magnetic pair breaking helps explain why some elements do not follow a naive density-of-states expectation.

In weak-coupling BCS theory, a useful scale isHere the phonon spectrum sets the attractive-interaction cutoff, $N(0)$ is the consistently defined Fermi-level density of states and $V_{\mathrm{eff}}$ the effective pairing attraction after repulsion is accounted for. Light hydrogen supports high vibrational frequencies; strong electron–phonon coupling and favourable metallic density of states can raise $T_c$. Pressure changes bonding, phase stability, electronic structure and phonon frequencies. It is not simply an independent multiplier in this expression. In a real strongly coupled hydride, Eliashberg theory uses the full spectral function and a screened Coulomb pseudopotential; the elementary exponential is a mechanism guide rather than a controlled quantitative fit to $203\mathrm{K}$.

Measure a four-terminal resistance that falls below the instrumental bound, and test that a magnetic field suppresses the transition. Independently detect diamagnetic screening and, where feasible, field-cooled flux expulsion or an AC susceptibility signal with controlled cell background. A resistance drop alone can reflect contact changes or a filamentary path; a consistent magnetic response is important evidence for superconductivity. A reproducible isotope shift upon replacing hydrogen by deuterium tests the involvement of lattice motion, although it is not by itself proof of superconductivity.

Extreme-pressure samples are microscopic and enclosed in a diamond-anvil cell. Reliable separate electrical contacts, homogeneous pressure, pressure calibration during cooling, chemical stability and the tiny magnetic signal relative to the anvils and gasket are practical difficulties. Structural identification matters because a sulfur-rich hydrogen compound formed under pressure need not retain the starting stoichiometry.

Historical correction: the original primary report associates the $203\mathrm{K}$ transition with pressures near $155\mathrm{GPa}$, while roughly $90\mathrm{GPa}$ is the metallization scale, not that same transition point. The printed pairing of $203\mathrm{K}$ with $90\mathrm{GPa}$ should not be treated as an established measured pair. The statement about the highest conventional transition temperature is also a historical statement in this 2018 examination.^[ref]

Use a normal-metal--insulator--superconductor junction with a weak tunnelling barrier. For a nearly constant normal-metal density of states, the differential conductance probes the thermally broadened superconducting quasiparticle density of states. At sufficiently low temperature, the ideal BCS theory result isThere is very little single-electron tunnelling for $|eV|<\Delta$, while a sharp onset occurs at $|eV|=\Delta$. The gap can therefore be obtained from the two symmetric onset voltages, $\Delta =e|V_{\mathrm{onset}}|$. For this one-superconductor junction the threshold is $\Delta /e$, not $2\Delta /e$.

Integrating at zero temperature gives the useful current sketchThe curve is odd, nearly flat inside the gap, and approaches the normal-state line $V/R_N$ at large $|V|$. The ideal gap-edge singularity is in the slope, not in the current itself. At finite temperature below the transition the onset rounds and thermally activated subgap current appears. Measuring several temperatures distinguishes this rounding from the closing of $\Delta (T)$.

The tree-level Cooper-pair scattering amplitude is simply $V$. Define the density of states at the Fermi surface, in the normalization of the question, byIn the one-loop Cooper diagram the two internal fermions have opposite momenta. After integrating over internal frequency, the remaining normal-energy integral containsConsequentlyand summing the leading geometric series gives the running BCS theory couplingFor repulsive $V>0$ it flows logarithmically toward zero. For attractive $V<0$, its denominator vanishes at the Cooper instability scaleAt this scale the normal Fermi liquid becomes unstable: opposite-momentum fermions form Cooper pairs, a superconducting or superfluid condensate develops, and a gap opens.

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.