Experiments on hollow superconductors show that the magnetic flux threading the hole is quantized in units of Φ₀ = ch/2e. Schwartz and Cooper (Phys. Rev. 137, A829 (1965)) traced this quantization not, as usually assumed, to the flux-dependent kinetic energy of the supercurrent, but to the pair-correlation energy term in the Gibbs free energy, whose minima occur when all Cooper pairs carry the same total angular momentum. Their analysis implies that quantization can fail: if the electron-electron interaction lacks rotational invariance, so that Cooper pairs may form between states of unequal angular momentum ("generalized pairing"), then near the half-integer flux points where the normal state switches fluxoid winding, the magnetic energy released by admitting a nonintegral flux can exceed the correlation energy lost through nonuniform pairing, and the cylinder settles into an intermediate-flux state. Schwartz and Cooper estimated the required pair-scattering ratio V₁/V₀ at roughly 0.6–1 and noted that a two-metal cylinder may reach ≈ 0.64, proposing eutectic normal/superconducting composites with small inner radius and low critical field as candidate specimens; they further predicted spontaneous equilibrium currents (flux at zero applied field, degenerate with its time reverse) in superconducting rings no larger than the coherence length with a single coherent Fermi surface. Six decades later, no realization of the intermediate-flux regime and no test of the coherent-ring prediction have been reported, while all known departures from Φ₀-periodicity in rings — h/e, half-quantum, and h/4e–h/6e effects — arise from pairing symmetry, nodal or multiband structure, higher-charge condensates, or spin-orbit coupling rather than from the Schwartz–Cooper mechanism. The open problem is whether the predicted breakdown of flux quantization can be realized or excluded: construct (or microscopically analyze) a globally superconducting, non-rotationally-invariant specimen in which the magnetic term dominates the pair-correlation term near the flux-jump points, and determine whether the equilibrium flux there becomes nonintegral; the source's related prediction of spontaneous currents in sub-coherence-length coherent rings is a separately conditioned secondary consequence.

Background

When a superconductor is multiply connected (a ring or hollow cylinder), the magnetic flux Φ threading the hole is constrained to an integer number n of flux quanta Φ₀ = ch/2e ≈ 2.07 × 10⁻¹⁵ Wb, where c is the speed of light, h is Planck's constant, and e is the electron charge; the factor 2e reflects the charge of a Cooper pair, the bound electron-electron pair of BCS (Bardeen–Cooper–Schrieffer) theory. Flux quantization was established experimentally in 1961 (Deaver–Fairbank; Doll–Näbauer). The modern textbook form is fluxoid quantization: the sum of the enclosed flux and the supercurrent's kinetic phase winding equals nΦ₀. In the Little–Parks experiment the superconducting transition temperature of a thin-walled cylinder oscillates with applied field with period Φ₀ because different fluxoid windings cost different kinetic energy. Byers and Yang framed the theory in terms of two classes of superconducting wave functions with free-energy minima at even and odd multiples of Φ₀, exchanged as the field sweeps through half-integer multiples.

Schwartz and Cooper (1965) decomposed the Gibbs free energy per unit volume of a superconducting cylinder into a magnetic term (minimized when the flux through the hole equals the applied flux), a kinetic term (which they showed is flux-independent once averaged over the two BCS pairing families — "even-line" and "odd-line" pairings of opposite total angular momentum), and a pair-correlation term Wᵥ, the condensation energy of the Cooper pairs, which is periodic in the enclosed flux and minimized when all pairs share the same total angular momentum. On this reading, flux quantization is a property of the pairing energy, not of the kinetic energy — a conclusion reached independently much later by Park and Kim (2004), who explicitly credited Schwartz and Cooper's generalized-pairing result.

Generalized pairing denotes pairing between single-electron states of unequal angular momentum, with amplitude governed by the pair-scattering matrix element V (subscript l−l′) between angular-momentum channels l and l′. In a rotationally invariant system this matrix element is diagonal — only V₀, which conserves the pair's total angular momentum, is nonzero; in a system lacking rotational invariance — their example is a cylinder made of two different metals, and they propose eutectics (mixtures of normal and superconducting regions that nevertheless percolate into a globally superconducting specimen, with the caveat that the entire cylinder must remain superconducting) — the off-diagonal element V₁, which mixes pair states whose total angular momenta differ by one unit, is nonzero. Their inequality (6.4) gives the critical ratio V₁/V₀ (0.61–1.00 for the parameters of their Table I) above which, near the "normal flux jump points" η = ±1/2, ±3/2 (in units of Φ₀, where the normal state changes its fluxoid winding), nonuniform pairing plus a nonintegral enclosed flux beats uniform pairing plus quantized flux. They estimate V₁/V₀ ≈ 0.64 for a two-metal cylinder — assuming both metals have the same Fermi energy and effective mass — tantalizingly close to the critical range. Their suggested experimental test is flux-quantization measurement on nonuniform samples with small inner radii, with attention to the behavior about those normal flux-jump points. They also note that generalized pairing undercuts the usual permanence argument for persistent currents, and they predict spontaneous equilibrium currents — a nonzero flux in zero applied field, degenerate with its time-reversed partner — in rings with no dimension exceeding the coherence length ξ (the scale over which the superconducting order parameter varies, typically of order 10⁻⁴ cm in conventional superconductors), provided the ring encloses a single coherent Fermi surface — that is, no specimen dimension exceeds ξ, so the whole specimen hosts one Fermi sphere rather than the many independent Fermi spheres over which the even/odd-line pairing average runs in long cylinders — and provided the pair-correlation term is not strongly flux-dependent.

The modern literature contains many departures from Φ₀ periodicity in superconducting loops, but each is mechanically distinct from the Schwartz–Cooper tradeoff: h/e periodicity in loops of nodal d-wave superconductors, whose superconducting energy gap vanishes along lines of the Fermi surface (Loder et al. 2008; Zhu 2008); field- or current-induced crossovers from h/2e to h/e periodicity (Sharon et al. 2018); half-quantum flux from π phase shifts at tricrystal grain boundaries in cuprate (copper-oxide) superconductors (Tsuei et al. 1994), from Ising pairing, in which spin–orbit coupling locks the electron spin to the lattice plane, in β-Bi₂Pd rings (Li et al. 2019), and from coexisting condensates with different center-of-mass momenta (Loder et al. 2013); charge-4e and charge-6e condensates, bound states of four or six electrons rather than two, in the kagome superconductor CsV₃Sb₅, which is built on the kagome lattice of corner-sharing triangles (Ge et al. 2024; Wang et al. 2025; Zhang et al. 2024); and spontaneous currents from spin-orbit coupling (Robinson et al. 2019) or from inhomogeneity and granularity (Ghinovker et al. 2001). None of these manipulates the angular-momentum structure of the pair scattering itself, and none addresses whether a magnetically soft, non-rotationally-invariant, globally superconducting specimen can sit at a nonintegral equilibrium flux. Conversely, Porforough (2025) has argued, from a supersymmetric Dirac-monopole model, that the Cooper-pair factor 2e in the Aharonov–Bohm phase is universal and can be recaptured without appeal to attractive pairing; that argument does not analyze the nonuniform angular-momentum pairing mechanism, and so neither constrains nor closes the Schwartz–Cooper question.

Problem Statement

Construct, or microscopically establish the impossibility of, a globally superconducting specimen that violates flux quantization through the Schwartz–Cooper mechanism: a non-rotationally-invariant specimen in which the magnetic term of the Gibbs free energy dominates the pair-correlation term near the normal flux-jump points at half-integer multiples of Φ₀ = ch/2e (η = ±1/2, ±3/2), so that the equilibrium enclosed flux becomes nonintegral. The target regime is a globally superconducting, non-rotationally-invariant specimen with inner radius well below the magnetic penetration depth (the distance over which a superconductor screens out magnetic fields) and low critical field (the applied field above which superconductivity is destroyed), in which the off-diagonal pair-scattering ratio V₁/V₀ approaches the critical values of Schwartz and Cooper's inequality (6.4) (0.61–1.00 in their Table I) while the entire specimen remains superconducting. The question is whether the equilibrium flux near the normal flux-jump points becomes nonintegral, whether the intermediate-flux regime can be excluded, or whether the source's free-energy argument contains a specific error; the behavior of the equilibrium flux about those points is of particular interest. The source's related prediction of spontaneous equilibrium currents in rings with no dimension exceeding ξ enclosing a single coherent Fermi surface is a separately conditioned secondary consequence noted in the Background.

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