Superfluidity and Superconductivity

Superfluids can sustain coherent flow; superconductors also respond to electromagnetic fields through their charged condensate. To understand either, ask how the system responds when its phase varies in space. The energy cost of that variation is phase stiffness. It connects microscopic coherence to flow, magnetic screening and vortices, but it is distinct from the energy needed to bind or break a pair.

This chapter develops neutral fluids and charged paired states using descriptions suited to different scales. London theory describes magnetic response when the amplitude is effectively fixed; Ginzburg–Landau theory allows that amplitude to vary; microscopic BCS theory explains a weak-coupling paired state. Vortices, junctions and proximity structures introduce further spatial and dynamical questions.

Helpful background. Bose–Einstein condensation and off-diagonal long-range order explain macroscopic occupation and correlation. BCS mean-field theory supplies fermionic pairing notation. Start with whichever branch fits your physical system; the chapter does not require all three in advance.

Enter this chapter

Physical question	Start here
How do helium and neutral films carry superfluid flow?	Superfluidity: stiffness, normal and superfluid components, sound and critical flow.
How does a bulk superconductor screen a weak magnetic field?	London theory: penetration depth, Meissner response and fluxoid quantization.
How do the order parameter and fields vary near a continuous transition?	Ginzburg–Landau theory: coherence length, critical fields and vortex structure.
How does weak attraction produce a fermionic paired state?	BCS theory: the gap equation, quasiparticles, thermodynamics and spectroscopy.
Why can an applied field produce magnetic flux lines and electrical resistance?	Vortices: vortex energetics, pinning, motion and dissipation.
How does a weak link carry phase-dependent current?	The Josephson effect: current–phase relations, interference and driven dynamics.
How does a superconductor alter a neighboring normal metal?	The proximity effect: induced correlations, spatial profiles and spectral changes.
What can symmetry and measurements reveal about nonstandard pairing?	Unconventional superconductivity: representations, sign changes, nodes and competing interpretations.

For an interface-scattering calculation, continue to proximity and Andreev physics. Once pairing and band structure are understood, topological superconductors introduce bulk invariants and boundary modes. Dated assessments of disputed mechanisms and new materials are collected in Quantum Matter research.

Pairing, condensation and phase stiffness

Pairing concerns correlations between particles. A two-body bound state is a property of a few-particle Hamiltonian. Cooper pairing is a many-particle instability near a Fermi surface and need not require a vacuum bound state. Local pair correlations can also survive when fluctuations destroy coherence between distant regions. The Bardeen–Cooper–Schrieffer calculation provides a specific microscopic realization, rather than a definition covering every superconductor.

Condensation concerns a large eigenvalue of a reduced density matrix. For bosons, a one-body eigenvalue proportional to the particle number defines Bose–Einstein condensation. For paired fermions, the relevant long-range structure is in a two-body density matrix. Writing an order parameter with a chosen phase is a convenient description, but a finite number-conserving state can have the corresponding correlations without a nonzero anomalous expectation value.

Stiffness concerns a response. Impose a gentle phase twist through a sample and find the resulting free-energy cost. In a homogeneous isotropic neutral two-dimensional fluid, the long-wavelength form is

$$F[\phi]-F[0]=\frac{\Upsilon}{2}\int d^2r\,|\nabla\phi|^2.$$

Here $\phi$ is dimensionless and $\Upsilon$ has units of energy. This coefficient describes collective rigidity; it is not a condensate fraction. The coefficient relevant at long distances includes the effects of shorter-scale fluctuations.

These distinctions matter particularly in two dimensions. A short-range interacting neutral fluid can have a Berezinskii–Kosterlitz–Thouless phase with algebraically decaying correlations and nonzero stiffness at nonzero temperature, yet no thermodynamic one-body condensate. At the transition, the long-distance stiffness approaches

$$\Upsilon(T_{\rm BKT}^{-})=\frac{2k_BT_{\rm BKT}}{\pi}.$$

This is the Nelson–Kosterlitz jump. A finite film rounds the transition; interpreting its response requires size and time scales, as explained in low-dimensional quantum gases and finite-size effects.

Charged phase rigidity and magnetic screening

For a field of charge $q_*$ and phase $\phi$, the combination that enters the kinetic energy is

$$\mathbf p_s=\hbar\nabla\phi-q_*\mathbf A.$$

For an electron-pair field, $q_*=-2e$ with $e>0$. Under $\mathbf A\mapsto\mathbf A+\nabla\chi$ and $\phi\mapsto\phi+q_*\chi/\hbar$, the two extra terms cancel. Thus a spatial phase gradient alone does not determine a measurable current. Gauge-invariant momentum, fields and phase differences do. Local gauge freedom is a redundancy of the description, not a physical symmetry whose different gauge choices label distinct samples.

The electromagnetic response also depends on how a perturbation is applied. A spatially uniform electric field changing in time probes the optical or transport limit. A static, slowly varying transverse magnetic field probes equilibrium screening. Their zero-frequency and long-wavelength limits need not commute; Scalapino, White and Zhang give a precise current-response formulation.

A clean normal metal can therefore have a zero-frequency Drude delta function when its current cannot relax, without having a bulk Meissner effect. Conversely, vortex motion can give a superconductor finite resistance in an applied field. Pairing, low resistance and magnetic screening provide complementary information. London theory develops the screening calculation; the vortex page explains how moving phase defects generate voltage.

What a gap measurement tells you

In the simplest uniform isotropic BCS model, the pairing field $\Delta$ also sets the minimum positive quasiparticle energy. This coincidence is useful, but it does not persist in every material or geometry. An anisotropic gap can have nodes, multiple bands can have different gaps, and an interface can support subgap states. A tunneling spectrum weights accessible single-particle states; optics creates transitions selected by current matrix elements. Even a nominal $2\Delta$ pair-breaking threshold requires a nonzero transition matrix element. In a perfectly clean Galilean-invariant single-band continuum, a uniform field cannot produce ordinary finite-frequency absorption through a current that is conserved.

Measured spectra also contain temperature and instrumental broadening, matrix-element weights and sometimes spatial averaging. A rounded edge does not by itself identify a quasiparticle lifetime. See scanning tunneling spectroscopy and terahertz and infrared probes for the connection to measured signals.

A useful example is a normal metal next to a conventional superconductor. In a fixed-phase mean-field description, let

$$\begin{aligned} F(\mathbf r)&=\langle\psi_\downarrow(\mathbf r)\psi_\uparrow(\mathbf r)\rangle,\\ \Delta(\mathbf r)&=g(\mathbf r)F(\mathbf r), \end{aligned}$$

where $g$ is the effective attraction in a local pairing model, with the appropriate microscopic regularization. In the normal region one may set $g=0$ while the boundary induces $F\ne0$. The local self-consistent pair potential is then zero, even though pair correlations and the spectrum differ from those of an isolated normal metal. Whether a hard minigap appears depends on geometry, boundaries and coherence; it is not guaranteed by $F\ne0$ alone. Belzig and collaborators explain these distinctions using quasiclassical transport theory. Diffusive Usadel and ballistic descriptions apply in different scattering regimes, while inverse proximity can require solving for the superconducting region as well.

Exercises

1. A condensate and a critical velocity

A homogeneous ideal Bose gas has excitations $\epsilon_p=p^2/(2m)$. A weakly repulsive gas at the Bogoliubov level has

$$E_p=\sqrt{\epsilon_p(\epsilon_p+2gn)},\qquad g>0.$$

Find the Landau bound $v_L=\inf_{p>0}E_p/p$ in each model. Does this establish the actual measured critical velocity?

Solution

For the ideal gas, $\epsilon_p/p=p/(2m)$ tends to zero: condensation alone has not produced a positive Landau bound. For the repulsive gas,

$$\frac{E_p}{p}=\sqrt{\frac{gn}{m}+\frac{p^2}{4m^2}}, \qquad v_L=\sqrt{\frac{gn}{m}}.$$

Interactions produce a linear sound mode at small momentum. This calculation tests energetic emission of the stated excitations in the homogeneous model. Boundaries, vortex nucleation, impurities and finite observation times can reduce measured onset velocities; a positive bound is not a complete account of persistent flow.

2. Large finite occupation without condensation

For a homogeneous square two-dimensional fluid, suppose the one-body correlation behaves as $g_1(r)\propto r^{-\eta}$ at large separation, with $0<\eta<2$. Estimate how the zero-momentum occupation $N_0$ and its fraction scale with side length $L$ at fixed density.

Solution

Translation invariance makes the zero-momentum occupation proportional to the spatial integral of $g_1$:

$$N_0\propto\int^{L}r\,dr\,r^{-\eta}\propto L^{2-\eta}.$$

Since $N\propto L^2$, the fraction scales as $N_0/N\propto L^{-\eta}$ and vanishes. Finite samples can still have a conspicuous central momentum peak. At the BKT transition, $\eta=1/4$ asymptotically, apart from logarithmic corrections; the fraction decreases only slowly. Its disappearance does not imply that stiffness vanishes throughout the low-temperature phase.

3. A current that does not decay

An isolated translationally invariant normal metal has conserved current and a Drude delta function. Why does this not establish superconductivity, and what additional response distinguishes it?

Solution

The delta function reflects the absence of a relaxation mechanism for a uniform driven current. Superconducting magnetic rigidity instead involves the static transverse current response and an equilibrium Meissner screening length. Paramagnetic and diamagnetic contributions must both be included, with the appropriate order of limits. Ordinary orbital magnetic susceptibility can remain in a normal metal, but it does not produce the finite long-wavelength Meissner kernel of a bulk superconductor. A persistent-current observation alone does not replace this distinction.

References

• J. Bardeen, L. N. Cooper, and J. R. Schrieffer, “Theory of Superconductivity,” Physical Review 108, 1175–1204 (1957), doi:10.1103/PhysRev.108.1175.
• W. Belzig, F. K. Wilhelm, C. Bruder, G. Schön, and A. D. Zaikin, “Quasiclassical Green’s Function Approach to Mesoscopic Superconductivity,” Superlattices and Microstructures 25, 1251–1288 (1999), doi:10.1006/spmi.1999.0710.
• J. M. Kosterlitz and D. J. Thouless, “Ordering, Metastability and Phase Transitions in Two-Dimensional Systems,” Journal of Physics C 6, 1181–1203 (1973), doi:10.1088/0022-3719/6/7/010.
• A. J. Leggett, Quantum Liquids: Bose Condensation and Cooper Pairing in Condensed-Matter Systems, Oxford University Press, 2006.
• D. R. Nelson and J. M. Kosterlitz, “Universal Jump in the Superfluid Density of Two-Dimensional Superfluids,” Physical Review Letters 39, 1201–1205 (1977), doi:10.1103/PhysRevLett.39.1201.
• D. J. Scalapino, S. R. White, and S. Zhang, “Insulator, Metal, or Superconductor: The Criteria,” Physical Review B 47, 7995–8007 (1993), doi:10.1103/PhysRevB.47.7995.
• M. Tinkham, Introduction to Superconductivity, second edition, Dover Publications, 2004.