Magnetic translation

For a lattice vector $r$,acts in the symmetric gauge asIt commutes with $(p-eA{)}^2/(2m)+V(x)$ when $V(x+r)=V(x)$.

For constant $B$,

A magnetic flux quantum is the flux scaleassociated with a coherent degree of freedom of electric charge $q\ne 0$. Its Aharonov-Bohm phase on a loop enclosing magnetic flux $\Phi$ is $q\Phi /\hslash$; adding ${\Phi }_0$ changes that phase by one full turn.

For an electron, the magnitude of the charge is $e$, giving $h/e$. A conventional superconducting Cooper pair has charge magnitude $2e$, giving $h/(2e)$. The relevant charge must therefore be stated: a superconducting flux scale and a single-electron interference period differ by a factor of two.

In the magnetic translation algebra, translations along two lattice vectors commute when the enclosed unit-cell magnetic flux obeysThis is a condition on the algebra for the specified cell, not a claim that every externally applied magnetic flux must be quantized. The distinct magnetic flux spectrum of a quantum ring illustrates flux-periodic energy levels.

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.