Pairing, Superfluidity, and Superconductivity

Pair formation, phase rigidity, and electromagnetic screening are distinct ingredients of superconductivity. Starting from the Cooper instability, this chapter connects paired quasiparticles to collective modes, vortices, retarded interactions, and topological boundary states.

Enter this chapter

Begin with The Cooper Instability and Pairing Channels to derive why an attractive channel destabilizes a Fermi surface.

Helpful background. The Cooper Instability and Pairing Channels and Landau Fermi-Liquid Theory develop the definitions used by the chapter’s opening pages.

Topics

1. The Cooper Instability and Pairing Channels — Why is an arbitrarily weak attractive interaction unstable in the Cooper channel and how are pairing symmetries classified?
2. BCS Mean-Field Theory and the Gap Equation — How does a pairing-field saddle produce a gapped spectrum, coherence factors, a gap equation, and thermodynamics?
3. Nambu–Gor’kov Green Functions and Anomalous Propagators — How do Nambu–Gor’kov fields encode normal and anomalous propagation without turning gauge-dependent anomalous expectation values into observables?
4. Bogoliubov–de Gennes Theory in Inhomogeneous Systems — How does Bogoliubov–de Gennes theory describe spatially varying pairing, boundaries, vortices, and disorder while preserving particle–hole redundancy?
5. Phase, Amplitude, and Leggett Collective Modes — How do phase, amplitude, and multiband relative-phase modes emerge from fluctuations about a paired saddle?
6. Superfluid Order and Phase Stiffness — How are paired order, helicity modulus, phase stiffness, superfluid density, and neutral phase response defined without relying on a gauge-dependent anomalous average?
7. Gauge-Invariant Meissner Response and Superfluid Weight — How does gauge-invariant electromagnetic response establish the Meissner effect and superfluid weight in charged paired matter?
8. Vortices and Topological Defects in Paired Matter — How do phase winding, core structure, energetics, and dimensionality determine vortices and other defects in paired matter?
9. Phase Winding, Flux Quantization, and Josephson Effects — How do gauge-covariant phase winding, flux quantization, and weak-link dynamics produce Josephson relations?
10. Migdal’s Theorem and Vertex-Correction Validity — Which energy, momentum, coupling, and band-structure hierarchy suppresses electron–phonon vertex corrections in Migdal theory?
11. Eliashberg Equations and Retarded Pairing — How do frequency-dependent normal and anomalous self-energies determine retarded pairing in Eliashberg theory?
12. The BCS–BEC Crossover — How does fermion pairing evolve from overlapping Cooper pairs to composite bosons across resonant interactions?
13. Unconventional Pairing Symmetries and Diagnostics — How are unconventional pairing symmetries classified and falsified using nodes, phase structure, spin/orbital content, and symmetry-resolved observables?
14. Unconventional Pairing Mechanisms and Competing Evidence — Which evidence distinguishes spin-, charge-, phonon-, excitonic-, or other pairing mechanisms once symmetry, retardation, and competing orders are separated?
15. Topological BdG Superconductors and Boundary Modes — How do BdG symmetry class, bulk invariant, boundary conditions, and defects determine Majorana boundary or vortex modes?
16. Majorana Platforms and Evidence Standards — Which platform observables discriminate Majorana modes from Andreev, Kondo, disorder, class-D, and device-specific alternatives?

How the ideas fit together

The BCS saddle reorganizes fermions into particle–hole mixtures and opens a pairing gap. Collective phase fluctuations then determine rigidity, and coupling that phase consistently to electromagnetic sources produces the Meissner response Bardeen, Cooper, and Schrieffer 1957, §§ II–V.

This sequence explains why a spectral gap alone does not prove coherent superconductivity. It also separates a self-consistent gap equation from a microscopic pairing mechanism. Topological boundary predictions further require a specified symmetry class, bulk gap, and boundary realization; a local zero-bias peak is not a unique diagnostic.

The comparison below separates the assumptions and tests needed for these different conclusions.

Paired-matter claim test matrix

Claim or regime	Defining input	Observable or invariant	Necessary control	Decisive negative test or ceiling
Cooper instability	Attractive eigenvalue of the antisymmetrized Fermi-surface kernel	Diverging normal-state pair susceptibility and instability scale $E_*$, or mean-field $T_c$ within a saddle approximation	Density-of-states, cutoff, and channel normalization fixed	Pair breaking or competing flow cuts off the logarithm before the claimed scale
BCS paired saddle	Gap function, dispersion, interaction, and ensemble	$E_{\mathbf k}$, coherence factors, gap and number equations	Saddle stability and ultraviolet matching	Negative fluctuation mode, violated number constraint, or cutoff-dependent observable
Nambu or BdG spectrum	Pairing matrix, Nambu convention, and boundary conditions	Poles, local density of states, and particle–hole-related eigenpairs	No double counting; complete basis and converged geometry	The complete numerical BdG eigenspectrum lacks its $\mathcal C$-related $\pm E$ partner, finite-size splitting is unresolved, or a claimed observable is gauge dependent
Collective mode	Zero of the analytically continued fluctuation kernel	Phase, amplitude, or relative-phase pole and residue	Conserving vertices, continuum threshold, and damping included	Response maximum moves with background model or lies inside an unresolved continuum
Superfluid stiffness	Free-energy curvature under a twist	Pair-phase helicity modulus or static transverse phase response	Thermodynamic and static limits declared	Curvature vanishes after size extrapolation although a pairing gap remains
Meissner response	Gauge-invariant current kernel	Static transverse screening kernel and penetration depth; optical delta or missing-area weight only after translating the limit order	Set $\omega=0$ before $\mathbf q\to0$; retain contact and paramagnetic terms; enforce Ward and sum rules; separate ballistic weights	A normal-state kernel survives spuriously, spectral weight is unaccounted for, or the result changes under a gauge-consistent reformulation
Vortex, fluxoid, or Josephson response	Compact phase, pair charge, geometry, and weak-link model	Winding, fluxoid sectors, current–phase relation, and voltage–frequency relation	Core scale, screening, capacitance, dissipation, and parity relaxation stated	Phase slips, trapped flux, ordinary harmonics, or poisoning explain the signal
Migdal–Eliashberg regime	Retarded interaction spectrum and electronic structure	Frequency-dependent $Z$, $\phi$, gap, and thermodynamics	Vertex ratio, momentum structure, bandwidth, coupling, and Coulomb treatment controlled	Vertex or nonadiabatic corrections are not small, or inversion is nonunique
BCS–BEC crossover	Matched scattering data, density, and number equation	Chemical potential, pair size, and excitation spectrum; the condensation transition only after a fluctuation or stiffness calculation	Range and density parameters; pairing distinguished from condensation	Pseudogap or molecular population is promoted to phase coherence without stiffness
Unconventional symmetry	Antisymmetry, crystal irrep, and spin–orbital structure	Nodes, relative phase sign, spin response, and symmetry-resolved perturbations	Domains, matrix elements, disorder, and surface reconstruction modelled	A different irrep or accidental-node state fits the same joint data
Microscopic mechanism	Irreducible pairing kernel and competing channels	Momentum- and frequency-resolved predictions under controlled perturbations	Common likelihood, calibrated nuisance terms, and held-out observables	Only $T_c$ or gap symmetry is fitted, or mixed mechanisms remain viable
Topological BdG phase	Local ultraviolet-complete gapped BdG Hamiltonian, protecting symmetry, and an interface or defect to a phase with a different invariant	Bulk invariant and boundary or defect spectral flow	Gap, locality, interface, symmetry, and finite-size convergence	The relevant gap closes, the protecting symmetry is absent, or the state is removable by a local symmetry-preserving perturbation
Majorana platform	Calibrated device forward model and parity sector	Nonlocal end correlation, parity dynamics, fusion, or braiding operation	Parent gap, disorder, temperature, transfer function, device replication, and trivial comparators	Local zero-bias, tune-up, or minimal-chain parity signal remains reproducible by Andreev, Kondo, disorder, or ordinary dynamics

Where to go next

Continue to Quantum Impurities, Polarons, and Kondo Matter to identify the impurity Hilbert space and the bath to which it couples.

References

• Bardeen, J., Cooper, L. N., and Schrieffer, J. R. (1957). “Theory of superconductivity.” Physical Review 108, 1175–1204. doi:10.1103/PhysRev.108.1175.