Superconductivity and Superfluidity
Superconductivity and superfluidity are the two places where quantum mechanics stops being microscopic. A single complex order parameter acquires a definite phase over a laboratory-sized sample, and the consequences are visible to the naked eye: a current that circulates for years without measurable decay, a magnet floating above a cooled disc, helium creeping up the wall of its own vessel. This chapter develops the phenomenological electrodynamics of London and of Ginzburg and Landau, the microscopic pairing theory of Bardeen, Cooper and Schrieffer, the energy gap and its spectroscopic signatures, the Josephson effects, and the superfluid phases of helium-4 and helium-3 — macroscopic quantum phenomena and the measured signatures that establish them.
It sits at the end of Part XII because it needs everything before it. The statistics of Quantum Statistics and the indistinguishability of Identical Particles supply Bose–Einstein condensation; the electron gas and Fermi surface of Electrons in Solids: Band Theory supply the normal state that becomes unstable; the lattice vibrations of Phonons and Lattice Dynamics supply the attraction that pairs the electrons; and the symmetry-breaking and critical-exponent machinery of Phase Transitions and Critical Phenomena supplies the language for the transition itself. Looking forward, the same broken gauge symmetry gives the photon a mass inside a superconductor and is the condensed matter ancestor of the Higgs mechanism of Electroweak Unification and the Higgs Boson; neutron-star interiors are believed to be superfluid (Compact Stars and Relativistic Astrophysics); and the flux quantum and the Josephson constant now define the practical volt in the SI of Measurement, SI Units, and the Theory of Errors. The experimental record — from the mercury run of 1911 to modern tunnelling and metrology — is collected in Experiment: Superconductivity. The standard modern monographs are Tinkham for superconductivity [Tinkham:1996], Khalatnikov for helium-4 [Khalatnikov:1965] and Vollhardt and Wölfle for helium-3 [Vollhardt:1990].
Superconductivity and Superfluidity: all derivations of this chapter are pending.
Discovery and phenomenology
Vanishing resistance
[Reserved: Kamerlingh Onnes' observation that the resistance of mercury falls discontinuously to immeasurably small values near \(4.2\,\mathrm{K}\) [Onnes:1911]; the distinction between a small resistance and a zero one, settled only by persistent-current decay bounds; the resulting upper limit on the resistivity, some twenty orders of magnitude below copper, and why a purely resistive account already fails — the full data are in Experiment: Superconductivity.]
Cooled through a temperature characteristic of the material — about \(4.2\,\mathrm{K}\) for mercury — the electrical resistance of certain metals does not fall smoothly towards a residual value, as Electrons in Solids: Band Theory would have it, but drops by more than four orders of magnitude within a few \(\mathrm{mK}\) to a value below the resolution of the measurement [Onnes:1911]. The statement that the resistance is exactly zero is not read off that measurement, which can only set an upper limit; it is inferred from the persistence of a current circulating in a closed superconducting loop, whose decay time is bounded below at order \(10^{5}\,\mathrm{yr}\) [File:1963] — a resistivity limit some seventeen orders of magnitude below that of copper at the same temperature. The transition is sharp, reversible, and occurs at a temperature that depends on the material but not on the measuring current so long as that current is small.
Why the normal state cannot produce this: the temperature-independent residual resistivity that impurity scattering guarantees in any metal, so that a discontinuous drop to zero requires a new state rather than a limit of the old one; and the conversion of the persistent-current decay bound into a resistivity bound through the inductance of the loop
The Meissner–Ochsenfeld effect
[Reserved: the expulsion of magnetic flux from the interior of a superconductor cooled through its transition in an applied field, observed by Meissner and Ochsenfeld [Meissner:1933]; why this is not a consequence of perfect conductivity — a perfect conductor freezes the flux it had, a superconductor expels it, so the superconducting state is a thermodynamic state and not a history; \(B=0\) in the bulk as the defining property, perfect diamagnetism \(\chi=-1\), and the thermodynamic critical field \(H_{c}(T)\) obtained from the condensation energy.]
A specimen of tin or lead placed in a modest magnetic field and then cooled through its transition temperature expels the field from its interior: the field outside the sample rearranges as the flux is pushed out, and inside the bulk \(B=0\) [Meissner:1933]. The final state does not depend on the order of operations — cooling first and then applying the field gives the same interior field, namely none — so the superconducting state is a thermodynamic state of the metal and not a record of its history. This is the observation that a perfect conductor cannot reproduce. Perfect conductivity implies only that the flux already threading the sample cannot change; a perfect conductor cooled in a field would trap that field forever. Above a critical field \(H_{c}(T)\), which vanishes at \(T_{c}\) and grows roughly parabolically below it, superconductivity is destroyed and the flux returns.
Derivation. The distinction is captured by replacing Ohm's law with a constitutive relation between the supercurrent and the field itself. Fritz and Heinz London postulated [London:1935]
whereas the acceleration of free charges in a perfect conductor gives the same relation with \(\vect{B}\) replaced by \(\pp\vect{B}/\pp t\) — which is the whole of the difference. Take the curl of the magnetostatic Ampère law \(\nabla\times\vect{B}=\mu_{0}\vect{j}_{s}\) of The Maxwell Equations, use \(\nabla\times(\nabla\times\vect{B})=\nabla(\nabla\cdot\vect{B}) -\nabla^{2}\vect{B}\) together with \(\nabla\cdot\vect{B}=0\), and substitute Equation (130.1):
The only solution of Equation (130.2) bounded in the interior of a half-space is \(B(x)=B(0)\ee^{-x/\lambda_{L}}\): the field survives only within a penetration depth of the surface, typically \(\lambda_{L}\approx50\,\mathrm{nm}\), and vanishes in the bulk. Because Equation (130.2) constrains \(\vect{B}\) and not \(\pp\vect{B}/\pp t\), it admits no field-cooled solution with trapped flux, and the Meissner effect becomes a theorem rather than an extra assumption. What has been assumed is Equation (130.1), whose physical content is the rigidity of the many-body wavefunction against the perturbation of a vector potential; that rigidity is what the microscopic theory of Section 130.3.4 has to supply.
∎Critical temperature, field and current
[Reserved: the critical surface in the \((T,H,J)\) space; the empirical parabolic law \(H_{c}(T)\approx H_{c}(0)[1-(T/T_{c})^{2}]\); Silsbee's rule relating the critical current to the field it generates at the surface [Tinkham:1996]; the tabulated values for the elemental superconductors [Roberts:1976] and the reason no elemental \(T_{c}\) exceeds about \(9.3\,\mathrm{K}\) at ambient pressure.]
The two-fluid picture
[Reserved: the Gorter–Casimir two-fluid model [Gorter:1934], in which a temperature-dependent fraction of the electrons is condensed and carries current without entropy; the consequent thermodynamics of the transition — a second-order transition in zero field with a jump, not a latent heat, in the specific heat; the model as the classical scaffolding that London and then BCS replaced with a microscopic wavefunction.]
Phenomenological electrodynamics
The London equations
[Reserved: Fritz and Heinz London's replacement of Ohm's law by a constitutive relation between the supercurrent and the vector potential [London:1935]; the two London equations, the second of which, combined with Ampère's law from The Maxwell Equations, yields \(\nabla^{2}\vect{B}= \vect{B}/\lambda_{L}^{2}\) and hence exponential screening over the penetration depth \(\lambda_{L}=\sqrt{m/\mu_{0}n_{s}e^{2}}\) — the Meissner effect as a theorem rather than an observation; typical \(\lambda_{L}\approx50\,\mathrm{nm}\); the London rigidity of the wavefunction as the physical content, and the London moment in a rotating superconductor.]
Non-local response and the coherence length
[Reserved: Pippard's microwave measurements showing that the measured penetration depth in alloyed tin does not follow the local London prediction, and his non-local generalization with a coherence length \(\xi_{0}\) over which the current responds to the field [Pippard:1953]; the analogy to the anomalous skin effect; \(\xi_{0}\approx\hbar v_{F}/\pi\Delta\) as later derived from BCS, and the ratio \(\kappa=\lambda/\xi\) that decides the type of the superconductor.]
Ginzburg–Landau theory
[Reserved: the complex order parameter \(\psi\) and the free energy expansion of Ginzburg and Landau [Ginzburg:1950], an application of the Landau theory of second-order transitions of Phase Transitions and Critical Phenomena with a gauge-covariant gradient term; the two Ginzburg–Landau equations; the identification \(\abs{\psi}^{2}=n_{s}\); surface energy between normal and superconducting regions and its sign change at \(\kappa=1/\sqrt{2}\); Gor'kov's derivation of the equations from BCS near \(T_{c}\), which fixed the charge in the gradient term at \(2e\) [Gorkov:1959].]
Type II superconductors and the Abrikosov lattice
[Reserved: Abrikosov's prediction that for \(\kappa>1/\sqrt{2}\) flux enters as a triangular lattice of quantized vortices between \(H_{c1}\) and \(H_{c2}\) rather than destroying the state at \(H_{c}\) [Abrikosov:1957]; the vortex core, the circulating supercurrent and the single flux quantum each vortex carries; the direct observation of the lattice by Bitter decoration [Essmann:1967]; flux pinning, critical currents and the practical superconducting magnet.]
Microscopic theory
The isotope effect
[Reserved: the simultaneous discovery by Maxwell [Maxwell:1950] and by Reynolds, Serin, Wright and Nesbitt [Reynolds:1950] that \(T_{c}\) of separated mercury isotopes scales as \(M^{-\alpha}\) with \(\alpha\approx1/2\); why an ionic mass in an electronic transition temperature is decisive evidence that the lattice mediates the interaction, since \(M^{-1/2}\) is the scaling of the phonon frequencies of Phonons and Lattice Dynamics; departures from \(\alpha=1/2\) in transition metals and their interpretation.]
Separated isotopes of mercury, chemically identical and differing only in the mass of the nucleus, have different superconducting transition temperatures, scaling as \(T_{c}\propto M^{-\alpha}\) with \(\alpha\) close to \(1/2\) — found simultaneously and independently by two groups [Maxwell:1950] [Reynolds:1950]. The result is decisive out of proportion to its size. Superconductivity is a property of the conduction electrons, and the ionic mass enters no electronic energy scale; but it is exactly the mass that sets the phonon frequencies of Phonons and Lattice Dynamics, which scale as \(M^{-1/2}\). The lattice must therefore mediate the interaction that produces the state. Departures from \(\alpha=1/2\) are measured in several transition metals and are informative in their turn.
Why an electron–phonon-mediated pairing energy carries the phonon frequency as its prefactor, so that the transition temperature inherits the ionic mass dependence; and the corrections — from the screened Coulomb repulsion between the electrons, and from strong coupling — that reduce the exponent below one half without removing the effect
The energy gap
[Reserved: the exponentially small electronic specific heat at low temperature as the first sign of a gap [Corak:1954]; the threshold in far-infrared absorption measured by Glover and Tinkham, which located the gap at \(2\Delta\approx3.5\,k_{B}T_{c}\) [Glover:1957]; the same gap seen directly in the tunnelling current–voltage characteristic [Giaever:1960]; the gap as an excitation gap for quasiparticles, not a gap in the single-particle band structure of Electrons in Solids: Band Theory.]
Three unrelated measurements agree that a superconductor cannot be excited at arbitrarily small energy. Its electronic heat capacity falls below \(T_{c}\) not as the linear \(\gamma T\) of a normal metal (Quantum Statistics) but exponentially, as \(\ee^{-\Delta/k_{B}T}\) [Corak:1954]. Far-infrared radiation is transmitted without absorption below a threshold photon energy and absorbed above it [Glover:1957]. And the current–voltage characteristic of a tunnel junction between a superconductor and a normal metal is flat until a threshold voltage, above which it turns sharply upward, so that \(\dd I/\dd V\) maps the quasiparticle density of states directly [Giaever:1960]. All three return the same gap, close to
a ratio that is roughly the same across the elemental superconductors. The gap is an excitation gap in a metal whose band structure has none: the states are there, and it is the correlated state that is expensive to disturb.
The gap as the energy to break one pair, obtained from the self-consistent gap equation of the pairing theory; the quasiparticle dispersion, whose minimum is the gap and whose density of states diverges at the gap edge, giving the observed tunnelling threshold and infrared edge; and the exponential activation of the heat capacity, together with the derivation of the measured ratio of twice the gap to the transition temperature as a parameter-free consequence of weak coupling
Cooper pairing
[Reserved: Cooper's theorem — two electrons added above a filled Fermi sea form a bound state for arbitrarily weak attraction, because the density of states at the Fermi surface is finite [Cooper:1956]; the logarithmic singularity and the non-perturbative binding energy \(\sim\hbar\omega_{D}\exp(-1/N(0)V)\), which no order of perturbation theory reproduces; the Fröhlich phonon-exchange interaction as the source of the attraction [Froehlich:1950]; the pair as an object of size \(100\,\mathrm{nm}\), overlapping many others, and so not a molecule.]
The BCS ground state
[Reserved: the variational pair-condensate wavefunction of Bardeen, Cooper and Schrieffer [Bardeen:1957]; the gap equation and its self-consistent solution; the Bogoliubov quasiparticle spectrum \(E_{k}=\sqrt{\epsilon_{k}^{2}+\Delta^{2}}\); the parameter-free predictions — \(2\Delta(0)=3.53\,k_{B}T_{c}\), the specific-heat jump \(\Delta C/\gamma T_{c}=1.43\), the isotope exponent \(1/2\) — and the coherence factors that make ultrasonic attenuation fall while nuclear spin relaxation first rises [Hebel:1959] [Morse:1957], a sign difference that is among the sharpest confirmations of the theory.]
Broken gauge symmetry
[Reserved: the superconducting state as spontaneously broken \(\U(1)\) symmetry; Nambu's demonstration that gauge invariance is compatible with the gap and the collective modes it requires [Nambu:1960]; Anderson's observation that the would-be Goldstone mode is absorbed by the electromagnetic field, leaving a massive photon — the Meissner effect read as a photon mass \(m_{\gamma}=\hbar/\lambda_{L}c\) [Anderson:1963] — and the direct line from there to the Higgs mechanism of Electroweak Unification and the Higgs Boson.]
Macroscopic quantum coherence
Off-diagonal long-range order
[Reserved: Penrose and Onsager's criterion for condensation in an interacting system — a macroscopic eigenvalue of the one-particle density matrix [Penrose:1956]; Yang's extension to the two-particle density matrix, which is what condenses in a superconductor, and his identification of off-diagonal long-range order as the single property shared by superfluid helium and every superconductor [Yang:1962]; the order parameter as an expectation value of a field operator, connecting to Open Quantum Systems and Decoherence.]
Flux quantization
[Reserved: single-valuedness of the condensate phase around a ring forces the enclosed flux to be an integer multiple of \(\Phi_{0}=h/2e=2.067833848\times 10^{-15}\,\mathrm{Wb}\); the independent measurements of Deaver and Fairbank [Deaver:1961] and of Doll and Näbauer [Doll:1961] in 1961, whose value of \(\Phi_{0}\) was half the naive \(h/e\) and thereby measured the carrier charge as \(2e\) — pairing established without any appeal to BCS; the Little–Parks oscillations of \(T_{c}\) with enclosed flux as the same quantization seen in a transition temperature [Little:1962].]
The magnetic flux trapped by a hollow superconducting cylinder is not continuous: it takes only integer multiples of a fixed quantum, and the measured quantum is \(\Phi_{0}=2.067833848\times 10^{-15}\,\mathrm{Wb}\) [Deaver:1961] [Doll:1961] [Mohr:2025]. The value is half the naive \(h/e\). Since the quantization argument fixes the quantum as \(h\) divided by the charge of the entity whose phase must be single-valued, the measurement determines that charge to be \(2e\) — pairing, established in 1961 by two independent groups, from a measurement that makes no appeal to any microscopic theory. The same periodicity in the enclosed flux appears in the transition temperature of a thin-walled cylinder [Little:1962].
Derivation. Let the superconducting state be described by a single complex field \(\psi=\abs{\psi}\,\ee^{\ii\theta}\) for carriers of charge \(q\) and mass \(m^{*}\). The gauge-invariant current is
Take a contour \(C\) running round the hole of the cylinder but lying deep inside the superconducting material, at a depth of many penetration lengths, where by Equation (130.2) the field and hence the current vanish. Setting \(\vect{j}_{s}=0\) in Equation (130.4) gives \(\hbar\nabla\theta=q\vect{A}\) along \(C\), and integrating round it,
where \(\Phi\) is the total flux enclosed. Single-valuedness of \(\psi\) forces the left-hand side to be \(2\pi\hbar n\) with \(n\) an integer, so \(\Phi=nh/q\). Nothing else enters: not the geometry, not the material, not the temperature. Reading the measurement backwards, \(q=h/\Phi_{0}=3.204353268\times 10^{-19}\,\mathrm{C}\), which is \(2e\) to the accuracy of the experiment. The argument is the same one that quantizes circulation in a neutral superfluid (Phenomenon 130.8); the charge is what turns a circulation into a flux.
∎The Josephson effects
[Reserved: Josephson's prediction that a supercurrent \(I=I_{c}\sin\varphi\) flows across a thin insulating barrier at zero voltage, and that a constant voltage drives the phase at \(\dd\varphi/\dd t=2eV/\hbar\), giving an alternating current of frequency \(2eV/h\) [Josephson:1962]; the observation of the dc effect by Anderson and Rowell [Anderson:1963a]; Shapiro's microwave-induced constant-voltage steps as the observation of the ac effect [Shapiro:1963]; the Fraunhofer dependence of \(I_{c}\) on applied flux.]
Two superconductors separated by an insulating barrier a few nanometres thick pass a current at zero voltage, up to a critical value \(I_{c}\) [Anderson:1963a]. That the barrier is not simply shorted is shown by the behaviour of \(I_{c}\) in a magnetic field: it does not fall monotonically but oscillates, vanishing whenever the flux threading the junction is an integer multiple of the quantum of Phenomenon 130.5, in the pattern of a single-slit diffraction envelope. Held at a constant voltage \(V\) instead, the junction carries an alternating current of frequency \(2eV/h\); irradiated with microwaves of frequency \(f\), its characteristic develops steps of constant voltage spaced by \(hf/2e\) [Shapiro:1963], which is how the alternating current is observed. Both effects were predicted before they were seen [Josephson:1962]. A frequency and a voltage are thereby locked together by fundamental constants alone, which is the basis of the practical volt.
Two weakly coupled condensates described by their amplitudes and phases: the coupled equations of motion for the pair, in which the current across the barrier is proportional to the sine of the phase difference and the phase difference evolves at a rate proportional to the electrochemical potential difference; the resulting zero-voltage supercurrent and the alternating current at the Josephson frequency; and the spatial variation of the phase difference across a junction in a field, whose integral gives the diffraction-like dependence of the critical current on the enclosed flux
SQUIDs and quantum metrology
[Reserved: two junctions in a ring interfere, and the critical current is periodic in the enclosed flux with period \(\Phi_{0}\) — the quantum interference observed by Jaklevic, Lambe, Silver and Mercereau [Jaklevic:1964]; the superconducting quantum interference device as the most sensitive magnetometer built, with a noise floor of a few \(\mathrm{fT}\) in a one-hertz band [Clarke:2004]; the Josephson constant \(K_{J}=2e/h=483597.8484\,\mathrm{GHz}/\mathrm{V}\) as an exact number in the revised SI [BIPM:2019] [Mohr:2025], and hence the practical realization of the volt.]
Superconductors beyond the BCS regime
The cuprates
[Reserved: Bednorz and Müller's report of a superconducting onset near \(35\,\mathrm{K}\) in the Ba–La–Cu–O system [Bednorz:1986], beyond any then-known \(T_{c}\); the rapid confirmation and the jump above the boiling point of nitrogen at \(93\,\mathrm{K}\) in Y–Ba–Cu–O [Wu:1987]; the layered copper-oxide structure, the doped Mott insulator as the parent compound, and the phase diagram with its pseudogap and strange-metal regions.]
Pairing symmetry in the cuprates
[Reserved: the phase-sensitive experiments that established \(d_{x^{2}-y^{2}}\) pairing — the corner SQUID of Wollman and co-workers [Wollman:1993] and the tricrystal ring experiment of Tsuei and co-workers [Tsuei:1994], which trapped a half-integer flux quantum where the order parameter changes sign; nodal quasiparticles and the resulting power-law rather than exponential low-temperature thermodynamics.]
Other families
[Reserved: magnesium diboride at \(39\,\mathrm{K}\), a two-gap phonon-mediated superconductor [Nagamatsu:2001]; the iron-based layered pnictides [Kamihara:2008]; the high-pressure hydrides, where conventional phonon-mediated pairing reaches \(203\,\mathrm{K}\) in sulfur hydride near \(155\,\mathrm{GPa}\) [Drozdov:2015] and higher still in lanthanum superhydride [Somayazulu:2019]; heavy-fermion and organic superconductors; the honest caveat that claims of near-ambient superconductivity have repeatedly failed replication.]
What remains unexplained
[Reserved: an explicit statement of what is established — that the cuprates are superconductors, that their carriers are pairs of charge \(2e\), that the order parameter has \(d\)-wave symmetry — and what is not: no mechanism for cuprate pairing commands consensus, and no theory predicts \(T_{c}\) for a given compound. The problem is listed among the open ones in What We Observe but Do Not Understand; the renormalization group of The Renormalization Group explains why a strongly correlated normal state resists the weak-coupling treatment that worked for lead.]
Superfluid helium-4
The lambda transition and the discovery of superflow
[Reserved: the specific-heat anomaly at \(2.17\,\mathrm{K}\) that gives the transition its name and separates He\,I from He\,II; the simultaneous reports of frictionless flow through narrow channels by Kapitza [Kapitza:1938] and by Allen and Misener [Allen:1938], published back to back; the fountain effect and the thermomechanical connection between heat and flow [Allen:1938b]; film creep; the modern microgravity measurement of the specific-heat exponent at the lambda point [Lipa:2003], one of the most precise critical exponents ever obtained, feeding Phase Transitions and Critical Phenomena.]
Liquid helium-4 cooled below \(2.17\,\mathrm{K}\) passes through channels and packed powders so narrow that the ordinary liquid would scarcely move at all, with no measurable pressure drop and hence no measurable viscosity — reported simultaneously by Kapitza and by Allen and Misener [Kapitza:1938] [Allen:1938]. The same liquid, measured by the drag on a rotating body, still exhibits a viscosity of the ordinary order, so the two experiments do not agree about what the viscosity is; the transition is marked independently by a specific-heat anomaly of a shape that gives it its name. Two further observations rule out any mundane explanation: helium-4 flows as a film up and over the wall of its container until the levels inside and outside are equal, and heating one side of a superleak drives a flow towards the heat rather than away from it, so that a jet of liquid can be raised by a lamp.
The two-fluid description in which the liquid behaves as an interpenetrating mixture of a normal component carrying all the entropy and all the viscosity and a superfluid component carrying neither, and the reconciliation of the capillary and rotating-body measurements that it provides; the thermomechanical effect as the statement that a temperature difference and a pressure difference are equivalent driving forces for the superfluid component; and Landau's criterion, in which an excitation spectrum with no low-lying free-particle branch forbids dissipation below a critical velocity — which is why an ideal Bose gas would condense without being a superfluid
Bose–Einstein condensation as the mechanism
[Reserved: London's identification of the lambda transition with the Bose–Einstein condensation of Quantum Statistics [London:1938], and the near-coincidence of the ideal-gas condensation temperature with the observed \(2.17\,\mathrm{K}\); why the identification is nevertheless not an explanation — the interacting liquid has a condensate fraction of only about \(7\,\mathrm{\%}\) at \(T=0\), so the superfluid fraction and the condensate fraction are different quantities; the contrast with the dilute atomic gases of Experiment: Bose–Einstein Condensation, where they nearly coincide; helium-4 as a boson, helium-3 as a fermion (Identical Particles).]
The two-fluid model
[Reserved: Tisza's proposal that He\,II behaves as an interpenetrating mixture of a normal viscous component and a superfluid component with zero entropy and zero viscosity [Tisza:1938]; Landau's reformulation in terms of the excitation spectrum, which supplies the normal density without invoking a condensate population [Landau:1941]; Andronikashvili's rotating stack of discs, which measured \(\rho_{n}(T)\) directly and confirmed the split [Andronikashvili:1946]; second sound as a counter-oscillation of the two components, predicted by Landau and observed by Peshkov [Peshkov:1944].]
The excitation spectrum and rotons
[Reserved: Landau's spectrum [Landau:1941] — a linear phonon branch at small momentum and a minimum, the roton, at \(p_{0}/\hbar\approx1.9\times 10^{10}\,/\mathrm{m}\) with gap \(\Delta/k_{B}\approx8.6\,\mathrm{K}\); the Landau criterion \(v_{c}=\min_{p}\varepsilon(p)/p\) for the breakdown of superflow, which is why an ideal Bose gas would not be a superfluid; Feynman's microscopic account of the spectrum and of vortex lines [Feynman:1955]; the spectrum measured directly by inelastic neutron scattering [Henshaw:1961].]
Quantized circulation and vortices
[Reserved: Onsager's proposal that circulation in a superfluid is quantized in units of \(h/m_{4}\approx 9.97\times 10^{-8}\,\mathrm{m}^{2}/\mathrm{s}\) [Onsager:1949a], and Feynman's development of the vortex-line picture [Feynman:1955]; Vinen's measurement of single quanta of circulation using the vibrating-wire method [Vinen:1961]; the rotating-bucket experiment and the vortex array; quantum turbulence as a tangle of such lines.]
The circulation trapped round a fine wire immersed in superfluid helium-4, measured through the splitting it induces between the two circular modes of the vibrating wire, does not take arbitrary values. It is found in integer multiples of
with \(m_{4}\) the mass of the helium-4 atom, and it changes between these values by whole steps and not continuously [Onsager:1949a] [Feynman:1955] [Vinen:1961]. Rotating a vessel of superfluid therefore does not set it turning as a body: the angular momentum is taken up by an array of discrete vortex lines. This is the neutral counterpart of Phenomenon 130.5, and the appearance of the bare atomic mass \(m_{4}\), rather than twice it, records that the condensing entity here is a single atom and not a pair.
Derivation. The argument is that of Phenomenon 130.5 with the charge set to zero. Writing the condensate field as \(\psi=\abs{\psi}\,\ee^{\ii\theta}\), the superfluid velocity is \(\vect{v}_{s}=(\hbar/m_{4})\nabla\theta\), so the flow is irrotational wherever \(\psi\neq0\). Round any closed contour on which \(\psi\) does not vanish, single-valuedness of \(\psi\) requires \(\theta\) to advance by \(2\pi n\), hence
The quantum in Equation (130.7) depends on the atomic mass and on Planck's constant and on nothing else, which is why the numerical value in Equation (130.6) can be quoted without reference to the temperature, the pressure or the apparatus. Two consequences are directly observed: circulation cannot decay smoothly, so a persistent flow is metastable against small perturbations; and non-zero circulation requires \(\psi\) to vanish along a line inside the contour, which is the vortex core.
∎Superfluid helium-3
Discovery of the superfluid phases
[Reserved: Osheroff, Richardson and Lee's observation of two anomalies on the melting curve of helium-3 near \(2.7\,\mathrm{mK}\) and \(2\,\mathrm{mK}\), initially attributed to the solid [Osheroff:1972]; Leggett's identification of them as transitions of the liquid into anisotropic superfluid phases with spin-triplet, \(p\)-wave Cooper pairs [Leggett:1972]; the A and B phases and the phase diagram in temperature, pressure and field.]
The order parameter and its consequences
[Reserved: a \(3\times3\) complex matrix order parameter and the broken relative spin–orbit symmetry, developed in [Vollhardt:1990]; the resulting textures, half-quantum vortices and nuclear-magnetic-resonance frequency shifts as the experimental handles; helium-3 as the laboratory in which unconventional pairing was first understood, and the template for the \(p\)-wave and \(d\)-wave superconductors of Section 130.5.2; Fermi-liquid theory as the necessary normal-state description.]
Where the phenomena reappear
[Reserved: a short closing survey. Neutron-star interiors as neutron superfluids and proton superconductors, invoked to explain pulsar glitches [Baym:1969] and taken up in Compact Stars and Relativistic Astrophysics; the superfluidity of dilute trapped atomic gases (Experiment: Bose–Einstein Condensation); the quantum Hall states of Experiment: The Quantum Hall Effect as a different macroscopic quantum order; and, in high-energy physics, the same broken-symmetry mathematics in Electroweak Unification and the Higgs Boson and in the chiral condensate of Quantum Chromodynamics. The common thread throughout is off-diagonal long-range order [Yang:1962], not any particular pairing mechanism.]