Current-loops, phase transitions, and the Higgs mechanism in Josephson-coupled multi-component superconductors

The $N$-component London $\mathrm{U}(1)$ superconductor is expressed in terms of integer-valued supercurrents. We show that the inclusion of inter-band Josephson couplings introduces monopoles in the current fields, which convert the phase transitions of the charge-neutral sector to crossovers. The monopoles only couple to the neutral sector, and leave the phase transition of the charged sector intact. The remnant non-critical fluctuations in the neutral sector influence the one remaining phase transition in the charged sector, and may alter this phase transition from a $3DXY$ inverted phase transition into a first-order phase transition depending on what the values of the gauge-charge and the inter-component Josephson coupling are. This preemptive effect becomes more pronounced with increasing number of components $N$, since the number of charge-neutral fluctuating modes that can influence the charged sector increases with $N$. We also calculate the gauge-field correlator, and by extension the Higgs mass, in terms of current-current correlators. We show that the onset of the Higgs-mass of the photon (Meissner-effect) is given in terms of a current-loop blowout associated with going into the superconducting state as the temperature of the system is lowered.