Magnetic translation algebra

For constant $B$,

Magnetic translations along two lattice vectors commute when their cell flux obeys

For a nonvanishing superconducting order parameter $\psi =\sqrt{n_s}{e}^{i\theta }$, $j=(q{n}_s/m)(\hslash \nabla \theta -qA)$. Single-valuedness gives phase winding $2\pi N$ around a closed curve, hence the fluxoid and flux quantization in a superconductor identity above. Magnetic flux alone is $Nh/q$ when the current integral vanishes. If the entire spanning surface is in a region where $D\psi =0$, the field vanishes there and the winding is zero; nonzero quantized flux requires a hole or a vortex core where that condition does not hold.

For uniform superconducting carrier density, the displayed combination of enclosed magnetic flux and the circulating supercurrent is quantized on a closed contour where the order parameter does not vanish. It follows by integrating the gauge-invariant phase gradient. Magnetic flux alone is quantized only when the current contribution on that contour is negligible.

The magnitude of one magnetic flux unit for an electron-pair order parameter, whose electric charge has magnitude $2e$. The enclosing contour must satisfy the flux condition in fluxoid and flux quantization in a superconductor; it is not a statement that every finite area inside the material contains an integer number of these units.