An order parameter organizes amplitude, gradients, and gauge response

Consider a single-component, isotropic Ginzburg–Landau functional near a continuous superconducting transition. Normalize Ψ so that its squared magnitude has units of effective pair density. With this convention α has energy units and β has energy times volume units. The gradient term uses an effective pair mass and the pair charge q=−2e. Other normalizations move factors among Ψ and the coefficients, so numerical coefficients cannot safely be compared across references without checking their definitions.

The local expansion contains the lowest even amplitude powers allowed by phase invariance, with positive β providing stability in this truncation. Near the mean-field transition α changes sign approximately linearly with temperature. Slowly varying fields justify retaining the leading squared covariant derivative. The functional is a coarse-grained approximation; strong fluctuations near criticality, large gradients, multiple competing components, or microscopic physics far below the transition can require additional terms or an entirely different description. Its effective density is not automatically an independently measurable count of tightly bound pairs.

Set the magnetic field and gradients to zero and minimize with respect to the nonnegative variable n=|Ψ|². For positive α, the minimum is n=0. For negative α, stationarity gives n=−α/β, which is positive because β is positive. Substitution lowers the free-energy density by α²/(2β). The amplitude therefore grows with the square root of the distance below the mean-field transition when α is linear in temperature. This square-root law is a mean-field result rather than a universal statement about critical exponents.

A spatially varying phase adds kinetic energy even when the magnitude remains constant, but only through the covariant combination of phase gradient and vector potential. Write Ψ as its magnitude times exp(iθ). The current is proportional to that combination multiplied by charge and amplitude squared. A nonzero gradient of θ alone is not enough to identify a current without specifying A. Likewise a nonzero vector potential can be physically harmless when compensated by phase variation in a region with zero magnetic induction.

Under A changing to A+∇χ, the phase changes to θ+qχ/ħ. These changes cancel inside the covariant momentum, leaving the magnetic field and current unchanged. Thus one should not describe the superconducting transition as literal spontaneous breaking of a local gauge redundancy. A chosen-gauge mean-field calculation often selects a complex phase, and relative phases enter interference experiments, but observable statements must be expressed through gauge-invariant correlations, currents, fields, or phase differences that include the vector-potential integral.

Use the vector lab only as a complex-amplitude picture. Reset and set both controls to angle zero degrees and magnitude one, predicting components (1,0). Reset and set angle 90 degrees and magnitude one, predicting (0,1). The magnitude stays one while its components rotate. This illustrates how changing a complex phase can leave the amplitude unchanged. The lab contains no vector potential and does not perform a gauge transformation or minimize a free energy; deciding whether a physical current changed requires information outside this geometric picture.