As an additional observation on the vector potential that follows from the above discussion concerns its dependence on the gauge. As we have discussed, the electronic vector potential can be transformed as in Eq. ( ) and it is, thus, not an observable quantity (as in classical electromagnetism). However, its curl is a robust physical observable and is a gauge-independent quantity: the curl of such an effective vector potential is related to the curl of the (physical) paramagnetic current of the system. Therefore, in Sec. III, we find it interesting to analyze the curl of the vector potential.