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

The $N$-component London $\mathrm{U}\left(1\right)$ superconductor is expressed in terms of integer-valued supercurrents. We show that the inclusion of interband 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 noncritical fluctuations in the neutral sector influence the one remaining phase transition in the charged sector, and may alter this phase transition from a $3\mathrm{D}XY$ inverted phase transition into a first-order phase transition depending on what the values of the gauge charge and the intercomponent 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.

Figure 1

A schematic phase diagram for the model with $N=2$. The top panel shows the case $\lambda =0$, while the lower panel shows the case with $\lambda >0$. Top panel, $\lambda =0$: Phase I is the fully symmetric normal phase with no superfluidity and no superconductivity. Phase III is the phase with no superconductivity, but nonzero superfluid stiffness in the neutral mode (metallic superfluid). Phase II is the low-temperature fully ordered state with finite Higgs mass in the charged sector and finite superfluid density in the neutral sector, a superconducting superfluid. The solid line separating phase I from phase II is a first-order phase transition line. The dotted line separating phase II from phase III is a critical line in the inverted $3\mathrm{D}XY$ universality class. The solid line separating phase I from phase III is a critical line in the $3\mathrm{D}XY$ universality class. Along the line $e=0$, we recover two uncoupled $3\mathrm{D}XY$ models, and the phase transition will be two superimposed independent phase transitions in the $3\mathrm{D}XY$ universality class. Bottom panel, $\lambda >0$: Phase ${\text{I}}^{\prime }$ is the high-temperature phase with no superconductivity. The entire phase is analytically connected with only a crossover regime separating the high-temperature phase from the lower-temperature phase. There is no spontaneous symmetry breaking in the neutral sector, since the Josephson coupling effectively acts as an explicit symmetry-breaking term in this sector, analogous to a magnetic field coupling linearly to $XY$ spins. Phase ${\text{II}}^{\prime }$ is the low-temperature superconducting state. The solid part of the line separating phase ${\text{I}}^{\prime }$ from phase ${\text{II}}^{\prime }$ is a first-order phase transition line. The dotted part is a critical line in the inverted $3\mathrm{D}XY$ universality class. For both $\lambda =0$ and $\lambda >0$, the line separating the superconducting states (II and ${\text{II}}^{\prime }$) from the nonsuperconducting state changes character from a first-order phase transition (solid line) to a second-order phase transition (dotted line) as via a tricritical point. The $3\mathrm{D}XY$ critical line separating phase I from phase III for $\lambda =0$ is converted to a crossover line in phase ${\text{I}}^{\prime }$ for $\lambda >0$. Along the $e=0$ line, the system is described by two neutral sectors coupled by an intercomponent Josephson coupling, such that the global $\text{U}\left(1\right)\times \text{U}\left(1\right)$ symmetry is reduced to a global $\text{U}\left(1\right)$ symmetry. Therefore, the phase transition reverts to a single $3\mathrm{D}XY$ transition.

Figure 2

Example of a current loop configuration in the case of strong interband Josephson coupling, when there is no screening of the charged-current interaction. Red and green lines represent currents of the individual fields ${\mathit{b}}_i$ flowing in the direction indicated by the arrows. Charged and neutral currents are therefore represented by overlapping red and green lines flowing either in the same or the opposite direction, respectively. The configuration shown represents a snapshot close to the charged transition, where a closed loop of charged current encircles pieces of a tightly bound composite neutral current. As there is no interaction between pieces of charged and neutral current, the inverted $3\mathrm{D}XY$ transition of the charged sector is not influenced by the tightly bound composite neutral currents.

Figure 3

Example of a current loop configuration in the case of weak, but nonzero, interband Josephson coupling. Red and green lines represent currents of the individual fields ${\mathit{b}}_i$ flowing in the direction indicated by the arrows. Charged and neutral currents are therefore represented by overlapping red and green lines flowing either in the same or the opposite direction, respectively. The present configuration shows the same loop of charged current encircling pieces of neutral current, as shown in Fig. 2. However, in the case of weak interband Josephson coupling, the individual currents will fluctuate away from the neutral-current configuration slightly close to the neutral crossover. The screening of the interaction between the segments of the outer charged composite current is accounted for by removing all tightly bound counterflowing currents in the interior, leaving only closed loops that change color an even number of times as the loops are traversed. These closed loops screen the charged-current interaction and may effectively change the sign of the interaction between the segments of charged currents, as explained in the text. This in turn may cause the transition of the charged sector to turn first order.