Phase winding quantizes a fluxoid before it quantizes flux alone

Where the order parameter is nonzero and smooth along a closed contour, its phase must return to itself modulo an integer multiple of 2π. The covariant momentum is m*v=ħ∇θ−qA, with q=−2e. Integrating around the contour relates the phase winding to the magnetic flux through any spanning surface and to the mechanical circulation. Expressing velocity through j=q|Ψ|²v gives the fluxoid formula when the effective density is approximately constant along the chosen contour.

With an oriented contour and integer phase winding n, the signed quantum is h/q, which is negative for our charge convention. It is convenient to relabel the winding integer as N=−n and define the positive flux quantum Φ₀=h/(2e). No physical sign is lost: contour orientation, current direction, and integer label must all be used consistently. If the density varies appreciably along the path, the mechanical-circulation form is safer than pulling a constant penetration depth outside the current integral.

For a contour deep within a thick ring, the current can be negligible and the fluxoid approximately reduces to magnetic flux alone. In a thin ring or a contour close to a screening region, the kinetic term can be substantial, so the flux need not equal an integer multiple by itself. The winding is protected only while the order parameter remains nonzero along a deforming contour. A phase slip or vortex crossing allows the amplitude to vanish locally and changes that topological sector.

An isolated singly quantized vortex has an amplitude-suppressed core on the coherence scale and a surrounding magnetic field on the penetration scale. Far outside the core and magnetic halo, its total flux approaches one Φ₀ in the usual bulk type-II setting. Between lower and upper critical fields, many such vortices can enter. Their repulsion in a clean isotropic sample often favors a triangular Abrikosov lattice; pinning, anisotropy, boundaries, and disorder can distort this ideal organization. A vortex array is a spatial superconducting texture, not a normal cylinder with an arbitrarily chosen magnetic flux.

Away from a neutral vortex core, phase winding produces an inverse-radius azimuthal velocity, so circulation is independent of contour radius. Charged vortices further include vector-potential screening. Classical rigid rotation instead has velocity proportional to radius and circulation proportional to enclosed area, with no integer restriction. Seeing arrows circulate is therefore insufficient evidence for a quantized vortex. The nonzero amplitude contour, gauge-covariant phase winding, and appropriate electromagnetic response are the decisive ingredients.

Use the fields lab as a deliberate classical comparison. Reset and set both controls to source strength zero and rotation Ω=1. At its displayed probe (1,0), predict Fx=0 and Fy=1. Reset and set source strength zero and Ω=2, predicting Fy=2. The numerical component doubles continuously. Arrow lengths are visually compressed, so use the Fx and Fy readouts for that factor. This softened planar source-plus-rotation illustration has neither an order parameter nor flux quantization; its smooth rotational field is not an Abrikosov vortex.