According to the standard theory of superconductivity, the origin of superconductivity is electron pairing. The induced current by a magnetic field is calculated by the linear response to the vector potential, and the supercurrent is identified as the dissipationless flow of the paired electrons, while single electrons flow with dissipation. This supercurrent description suffers from the following serious problems: (1) it contradicts the reversible superconducting-normal phase transition in a magnetic field observed in type I superconductors; (2) the gauge invariance of the supercurrent induced by a magnetic field requires the breakdown of the global [Formula: see text] gauge invariance, or the nonconservation of the particle number; and (3) the explanation of the ac Josephson effect is based on the boundary condition that is different from the real experimental one. We will show that above problems are resolved if the supercurrent is attributed to the collective mode arising from the Berry connection for many-body wavefunctions. Problem (1) is resolved by attributing the appearance and disappearance of the supercurrent to the abrupt appearance and disappearance of topologically protected loop currents produced by the Berry connection; problem (2) is resolved by assigning the non-conserved number to that for the particle number participating in the collective mode produced by the Berry connection; and problem (3) is resolved by identifying the relevant phase in the Josephson effect is that arising from the Berry connection, and using the modified Bogoliubov transformation that conserves the particle number. We argue that the required Berry connection arises from spin-twisting itinerant motion of electrons. For this motion to happen, the Rashba spin–orbit interaction has to be added to the Hamiltonian for superconducting systems. The collective mode from the Berry connections is stabilized by the pairing interaction that changes the number of particles participating in it; thus, the superconducting transition temperatures for some superconductors is given by the pairing energy gap formation temperature as explained in the BCS theory. The topologically protected loop currents in this case are generated as cyclotron motion of electrons that is quantized by the Berry connection even without an external magnetic field. We also explain a way to obtain the Berry connection from spin-twisting itinerant motion of electrons for a two-dimensional model where the on-site Coulomb repulsion is large and doped holes form small polarons. In this model, the electron pairing is not required for the stabilization of the collective mode, and the supercurrent is given as topologically protected spin-vortex-induced loop currents (SVILCs).
Josephson effect in type-I Weyl semimetals