Vortex topological charge: winding, fluxoid and magnetic flux

Scope

A superconducting vortex is a defect of the real-space order parameter. For a single complex component that stays nonzero on a closed contour $C$, its phase winding is $n=(2\pi)^{-1}\oint_C\nabla\phi\cdot d\mathbf l\in\mathbb Z$. The contour must avoid cores and other zeros. Reversing its orientation reverses the sign; enclosing several defects gives their net winding. This integer is distinct from the momentum-space Chern number of a band or Bogoliubov Hamiltonian. R1

Winding, fluxoid and measured magnetic flux answer different questions. Fluxoid combines magnetic flux with supercurrent circulation. A detector measures a geometry-dependent magnetic response, which need not equal the fluxoid around a chosen contour. Junction phase shifts and multicomponent textures require additional constraints; a fractional magnetic signal alone does not specify a new bulk condensate charge, Chern number or Majorana mode. R1 R6 R7

Mathematical assumptions

Use the explicit positive-charge phase convention $q=2e$, with $e\gt0$, and the local London relation $\mu_0\lambda_L^2\mathbf j_s=(\hbar/q)\nabla\phi-\mathbf A$. For constant $\lambda_L$ on the contour, integration gives $\Phi+\mu_0\lambda_L^2\oint_C\mathbf j_s\cdot d\mathbf l=n\Phi_0$, where $\Phi_0=h/(2e)\approx2.0678\times10^{-15}$ Wb. With a negatively charged pair-field convention, the associated phase winding reverses sign. These are equivalent conventions when used consistently. The simple constant-$\lambda_L$ expression assumes a local, uniform condensate response away from the core. R1 R4

Only when the circulation term is negligible does magnetic flux itself approach $n\Phi_0$. A lumped uniform ring makes the distinction explicit: $I=(n\Phi_0-\Phi_{\rm ext})/(L_g+L_k)$, $\Phi=\Phi_{\rm ext}+L_gI$, and $\Phi+L_kI=n\Phi_0$. Here $L_g$ and $L_k$ are geometric and kinetic inductance. At zero applied flux with $n=1$ and $L_k=9L_g$, the magnetic flux is only $0.1\Phi_0$ although the fluxoid is one quantum. Twelve illustrative branches check this identity and the energy $(n\Phi_0-\Phi_{\rm ext})^2/[2(L_g+L_k)]$; they do not model phase-slip switching or an experimental device. R1 R7

Twenty-four discrete phase-loop checks compare enclosed defect charges, contour reversal and smooth single-valued gauge changes at two resolutions. A contour intersecting a zero is rejected. These checks establish the numerical examples under the stated complex-field construction, not a classification of every superconducting order-parameter manifold. Singular gauge choices require separate treatment and were not tested. R1

For an ideal triangular array with one flux quantum per cell, $B(\sqrt3 a^2/2)=\Phi_0$ gives $a=\sqrt{2\Phi_0/(\sqrt3 B)}$. At mean induction $B=1$ T this is approximately $48.9$ nm. Four geometric checks verify the cell area; they do not establish that a particular sample has a triangular lattice. Use mean induction $B$, not applied field $H$ without a stated magnetization approximation. The later Abrikosov lecture favors the triangular array in the isotropic model but allows crystal-dependent alternatives; its result should not be attributed uncritically to the different early numerical estimate in the 1957 paper. R1 R5

Experimental criteria

Deaver and Fairbank reported discrete trapped flux in tin cylinders consistent with $hc/2e$ in Gaussian units, including a first-step uncertainty of approximately 20 percent and explicit geometric corrections. Doll and Näbauer independently observed steps in lead-tube measurements, but their original report compared approximately 0.2 Oe spacing with 0.5 Oe predicted using $hc/e$ and left that discrepancy unresolved. Preserve the observations and uncertainties of each paper instead of rewriting both reports as identical precision determinations. The SI flux quantum has no factor of $c$. R2 R3

Tsuei and colleagues reported spontaneous half-quantum flux in a three-junction YBCO tricrystal ring, absent in the two-junction controls, consistent with d-wave pairing. Their 4.2 K value is the scanning SQUID measurement temperature. Interpreting such a result requires the junction geometry and phase constraints, trapped-flux controls and a calibrated magnetic response; it is not an observation of an isolated half vortex in every ordinary single-component superconductor. The original abstract has been checked here, not the complete flux-map analysis. R6

Bauer and colleagues used niobium loops with PdNi ferromagnetic junctions and conventional-loop controls. Their final paper describes 2 K field sweeps, cooling-field reversal and subtraction of the Hall probe's temperature-dependent background; the low-temperature spontaneous flux approaches half a quantum. The final Figure 4 onset near 5 K replaces the author manuscript's approximate 5.5 K value. This is a device spontaneous-current scale, not a new bulk Nb transition temperature. Neither that onset nor the 2 K measurement is entered as experimental $T_c$. R7

Finite inductance matters even in a simple sinusoidal single-π-junction model. At zero applied field and negligible ring kinetic inductance, set $x=2\pi\Phi/\Phi_0$, $\beta_L=2\pi L_gI_c/\Phi_0$ with $I_c\gt0$, and normalized energy $u=x^2/2+\beta_L\cos x$. The stable nonzero solutions obey $x=\beta_L\sin x$ for $\beta_L\gt1$. Illustrative values $\beta_L=2,10,100$ give positive fluxes approximately $0.302,0.454,0.495$ quanta; at $0.5$ the minimum is zero. The half-quantum limit is approached with large $\beta_L$, not imposed exactly at every inductance. This independently checked model is distinct from the paper's gamma notation and from a multijunction tricrystal. R6 R7

Limits and disputes

Current-driven vortex motion can produce an electric field and dissipative transport. The checked Bardeen–Stephen abstract supports this result in a local vortex-motion model. Its full viscosity and Hall-response calculation has not been reproduced here. R8

The full Byers–Yang proof, complete 1994 tricrystal analysis, multicomponent fractional-vortex energetics and experimental raw-data audits remain open. Selected original scans, the later Abrikosov lecture and the published Bauer passages support the bounded discussion above. The local winding and ring calculations are illustrative checks. They do not complete scientific acceptance or establish a material's topological superconductivity. R1 R2 R3 R4 R6 R7

Discovery implications

Store the contour, phase convention, inductive terms, field history, probe calibration and sample state alongside a vortex claim. Separate cell flux inferred from average induction, local sensor flux, fluxoid and momentum-space invariants. For a proposed fractional defect, identify the order-parameter or Josephson constraint and test competing trapped-flux explanations. These distinctions make a magnetic signal useful for comparing hypotheses without treating it as a complete pairing or topology diagnosis. R1 R6 R7

Related topics

BKT transition, Chern number, Majorana zero modes, and geometric superfluid weight.

Material cases

Values describe individual reports, not a ranking. Unknown means the checked source did not supply the field at this review scope.