The association of broken symmetries with phase transitions is ubiquitous in condensed matter physics: crystals break translational symmetry, magnets break rotational symmetry, and superconductors break gauge symmetry. However, despite the frequency with which it is made, this last statement is a paradox. A gauge symmetry, in this case the U(1) gauge symmetry of electromagnetism, is a redundancy in our description of nature, so the notion of breaking such a “symmetry” is unphysical. Here, we will discuss how gauge symmetry breaks, and doesn't, inside a superconductor, and explore the fundamental relationship between gauge invariance and the striking phenomena observed in superconductors.
This is because the Lagrangian has a term coupling the particle to the field. The canonical momentum then has an additional term ∼ qA which must be subtracted out to get the “physical” momentum .
Since α(x, t) is dependent on space and time, derivatives acting on the gauge transformed wave-functions give , and similarly for time derivatives.
Since for any system of definite particle number the energy eigenstates will be simultaneous eigenstates of the number operator, there is no way the system can time evolve into a state with different particle number. Put another way, since the number operator commutes with the Hamiltonian, particle number must be conserved.
Even though the pairs in the condensate are comprised of electrons, we treat them as different objects. To be more precise, the electrons we speak of are quasiparticles, not bare electrons.
Expanding the free energy, which is known to be discontinuous at a phase transition, as an analytic series near the discontinuity can only be justified if we understand the free energy as the result of a saddle point approximation of the appropriate partition function, see Ref. 8.
Details of the Ginzburg-Landau theory, which we have glossed over, ensure that in the superconducting state r < 0, so the gap magnitude Δ†Δ = –r/u is a positive quantity, see Ref. 14.
Once could conceivably ask why the quadratic function must be centered about zero, i.e., why could we not have something like ? The effective action would then be minimized when , which we know from Eq. (55) represents current flow.
The Anderson-Higgs mechanism also generates the mass of quarks and electrons due to a Yukawa coupling that has no analogue in superconductors, and thus is not discussed here.